user177107 should be commended for asking this clever question.
In short, the described configuration is possible, at least in principle.
First of all, let us recall that the solar tides in a planet whose spin is synchronised with the star are always working to ensure descent, see eqn (150) in this work. On the other hand, the solar tides in a nonsynchronised planet are working to repel the planet from the star.
A large moon orbiting a not too large planet may synchronise this planet's rotation. Planet's rotation synchronised with the moon will not be synchronised with the star. As a result of this, the solar tides in the planet may be contributing either to the tidal ascent or to the tidal descent to the star, dependent on whether the planet's (to be exact, the planet-moon barycentre's) mean motion $n$ about the star is smaller or larger than the planet's spin rate. As demonstrated here the spin rate of a planet synchronised by a moon will, typically, be several times higher than the mean motion $n$ of the planet about the star. So the solar tides in the planet will be repulsive -- and this is one possible mechanism helping a close-in planet to avoid engulfment.
Two technical warnings to those who decide to integrate this problem.
(1) If a prograde moon starts above the synchronous radius (like Deimos or the Moon), it will usually be ascending. I am saying `usually', because different is a situation where the eccentricity is noticeable. (And it is likely that Phobos was at some point in this exceptional situation -- which helped it to fall down through the synchronous orbit.) Assuming that nothing prevents the orbit from circularisation, and that the moon is ascending, we have to take into account that at some point the moon may be lost to the star. Naively, this should happen when its semimajor axis crosses the Hill radius $r_H$. In reality, the orbit becomes unstable already at $0.49 r_H$, for a prograde-orbiting moon, and $0.93r_H$, for a retrograde-orbiting one. Keep this detail in mind if you try to integrate.
(2) If a prograde moon starts below the synchronous radius, it will be falling down and, at the same time, accelerating the planet's rotation. It may result in synchronisation of the planet with the moon -- provided this end state is attained before the descending moon crosses the Roche radius.