I imagine that a planet will become too hot for Earth type life - or even hypothetical life with different biochemistries might be able to live at much higher temperatures than Earth life - long before it becomes engulfed by its star's atmosphere.
Even the coolest stars have surface temperatures of a few thousand degrees, so a planet which is close enough to the surface of a star to be engulfed n the atmosphere of the star would probably be far too hot for even hypothetical alien biochemistries.
Since your planet is atmosphere less, there would be no e heat exchange by liquids or gases between the day side and the night side. But heat would spread through the rocks from the hot side to the cooler side and heat it up.
But I am unable to calculate what conditions would be necessary for the dark side of the planet to have a specific temperature.
I notice that you ask:
But at which point these effects will make the conditions on the dark side unbearable?
And I have to ask:
"Who or what the dark side conditions would be unbearable for".
Are you asking about human explorers landing in spaceships and building bases and walking around in spacesuits? Do you want to know how hot the surface can get before it becomes too hot for humans in spacesuits and air conditioned vehicles and air conditioned bases?
Or are you asking about what would make the dark side unbearably hot for native life forms of the planet?
On Earth, water exists in three different states, solid, liquid, and gas, and often transitions between different states. And we would expect that any other chemical used a a solvent and medium for alien life would also transition between solid, liquid, and gas at the temperatures suitable for that hypothetical alien life with radically different biochemistry.
As the atmospheric pressure drops, the temperature at which a liquid boils and also drops. At a low enough atmospheric pressure, that liquid will sublimate and transition directly from solid to vapor, with no liquid stage. In a vacuum or near vacuum, all substances will transition directly from solid to gaseous, and any liquid will quickly boil away.
And your planet is defined as:
an atmosphere-less planet, tidally locked to a sun-like star.
So your planet should not have any bodies of any liquid which life forms could use on its surface.
The only way to give your planet any native life is to give it large bodies of liquid on the dark side covered by very thick layers of frozen liquid.
So there could be inland seas of liquid methane, covered by thick sheets of frozen methane. And possibly there could be hypothetical lifeforms which use liquid methane in the methane ocean. As the heat from the day side is conducted through rocks to the night side, the liquid methane ocean will heat up, possibly becoming too hot for methane based life. And eventually the methane ocean will turn into methane vapour and possibly blow open the sheets of methane ice above it.
And maybe there are inland seas of liquid ammonia, covered by thick sheets of frozen ammonia. And possibly there could be hypothetical lifeforms which use liquid ammonia in the ammonia ocean. As the heat from the day side is conducted through rocks to the night side, the liquid ammonia ocean will heat up, possibly becoming too hot for ammonia based life. And eventually the ammonia ocean will turn into ammonia vapor and possibly blow open the sheets of ammonia ice above it.
And maybe there are inland seas of liquid water, covered by thick sheets of frozen water. And possibly there could be hypothetical lifeforms which use liquid water in the water ocean. As the heat from the day side is conducted through rocks to the night side, the liquid water ocean will heat up, possibly becoming too hot for water based life. And eventually the water ocean will turn into water vapor and possibly blow open the sheets of water ice above it.
And maybe there are inland seas of liquid sulfur, covered by thick sheets of frozen sulfur. And possibly there could be hypothetical lifeforms which use liquid sulfur in the sulfur ocean. As the heat from the day side is conducted through rocks to the night side, the liquid sulfur ocean will heat up, possibly becoming too hot for sulfur based life. And eventually the sulfur ocean will turn into sulfur vapor and possibly blow open the sheets of sulfur ice above it.
And of course different rock temperatures on the night side would be necessary for methane based life, or ammonia based life, or water based life, or sulfur based life. Or life using some other solvent.
I note that your planet is described as:
an atmosphere-less planet, tidally locked to a sun-like star
If the star has the same mass and luminosity as the Sun, the distance at which a planet in our solar system would become tidally locked would be the distance at which your imaginary planet would become tidally locked to its star. And thus the planetary temperature would be similar to that of a planet at that distance from the Sun.
Habitable Planets for Man Stephen H. Dole, 1964, 2007, discusses the various factors making worlds habitable or not for humans.
On page 71 table 9 lists the calculated strengths of tidal retardation effects on various objects in the Solar System and lists wherever they are tidally locked or not.
Dole concludes on page 70 that when the factor of h squared is somewhere between 1.2 and 2.0, the planet's rotation will be "stopped" (actually tidally locked to its primary).
However, there is a problem with Dole's conclusion. It was written when the planet Mercury, which is much closer to the Sun that Earth and thus much hotter on the day side, was believed to be tidally locked to the Sun, keeping one side eternally facing the Sun.
But in 1965 it was discovered that Mercury is not tidally locked to the Sun.
For many years it was thought that Mercury was synchronously tidally locked with the Sun, rotating once for each orbit and always keeping the same face directed towards the Sun, in the same way that the same side of the Moon always faces Earth. Radar observations in 1965 proved that the planet has a 3:2 spin-orbit resonance, rotating three times for every two revolutions around the Sun. The eccentricity of Mercury's orbit makes this resonance stable—at perihelion, when the solar tide is strongest, the Sun is nearly still in Mercury's sky.
The rare 3:2 resonant tidal locking is stabilized by the variance of the tidal force along Mercury's eccentric orbit, acting on a permanent dipole component of Mercury's mass distribution. In a circular orbit there is no such variance, so the only resonance stabilized in such an orbit is at 1:1 (e.g., Earth–Moon), when the tidal force, stretching a body along the "center-body" line, exerts a torque that aligns the body's axis of least inertia (the "longest" axis, and the axis of the aforementioned dipole) to point always at the center. However, with noticeable eccentricity, like that of Mercury's orbit, the tidal force has a maximum at perihelion and therefore stabilizes resonances, like 3:2, enforcing that the planet points its axis of least inertia roughly at the Sun when passing through perihelion.
The original reason astronomers thought it was synchronously locked was that, whenever Mercury was best placed for observation, it was always nearly at the same point in its 3:2 resonance, hence showing the same face. This is because, coincidentally, Mercury's rotation period is almost exactly half of its synodic period with respect to Earth. Due to Mercury's 3:2 spin-orbit resonance, a solar day lasts about 176 Earth days. A sidereal day (the period of rotation) lasts about 58.7 Earth days.
So it is possible that a planet would have to be even closer to a sun-like star, and thus even hotter, than Mercury to be tidally locked. But it is possible that the reasons why Mercury has a 3:2 spin-orbit resonance would not apply to all planets orbiting at Mercury's distance from sun-like stars, and that some planets could become tidally locked at Mercury's distance and even farther from a sun-like star.
The planets Mercury and Venus are not tidally locked to the Sun, and thus do not have eternal day on one side and eternal night on another side. But they do have very long days and nights.
On Venus, there is little difference in temperature between day and night, because the dense atmosphere spreads heat evenly around the planet.
Mercury has a very long day, but does have alternations of day and night.
The surface temperature of Mercury ranges from 100 to 700 K (−173 to 427 °C; −280 to 800 °F) at the most extreme places: 0°N, 0°W, or 180°W. It never rises above 180 K at the poles, due to the absence of an atmosphere and a steep temperature gradient between the equator and the poles. The subsolar point reaches about 700 K during perihelion (0°W or 180°W), but only 550 K at aphelion (90° or 270°W). On the dark side of the planet, temperatures average 110 K. The intensity of sunlight on Mercury's surface ranges between 4.59 and 10.61 times the solar constant (1,370 W·m−2).
A darkside temperature of 110 Kelvin would equal minus 261.67 degrees Fahrenheit (-163 °C).
So the temperature on a typical point of Mercury will rise and fall by hundreds of degrees every Mercurian day. Mercury rotates with respect to the stars (sidereal day) every 59 Earth days, but makes a full rotation with respect to the Sun (a solar day) every 175.97 Earth days.
So a point on Mercury which has a temperature of about 110 K will have it rise by at least 300 K during a period of 87.985 Earth days, and then fall again by at least 300 K in the next 87.985 Earth days.
I think this means that the rocks and other surface materials on Mercury do not conduct heat very well. And possibly that might mean that the dark side of a tidally locked airless planet might be very, very cold.
When it was believed that Mercury was tidally locked to the Sun, it was logical to assume that the dark side of Mercury was the coldest place in the solar system, and thus Larry Niven's story set on the dark side of mercury was titled "The Coldest Place".
Thus it is possible that a tidally locked airless planet would have to be very close to its star for enough heat from the day side to make it to the night side and raise the temperature to unbearably hot instead of just right or instead of unbearably cold.