You ask about the larger body's magnetosphere providing protection from the solar rwind for the smaller body. Presumably you want the magnetosphere to prevent the solar wind from stripping away the atmosphere of the smaller body.
You describe the smaller body:
The other body is 10.4M☽, or about .13M🜨, slightly larger than Mars and lacking a large iron core.
And possibly that world would be so small that it would lose atmopshere so fast that worrying about the solar wind would be pointless.
Wikipedia has an article about atmospheric escape which should be the first place to look to find out about different possible methods of atmospheric loss.
If you have any desire for either of your hypothetical planets to be habitable for oxygen brathing animals like humans in particular, or for liquid water using life in general, you should do research on planetary habitability.
There is one scientific study about the requirements for planets to be habitable for human beings from Earth (and thus for any lifeforms with the same requirements) Habitable Planets for Man, Stephen H. Dole, 1964.
On pages 34 to 35 Dole mentions complex equations for how fast a world's atmopshere will escape into space. And also mentions a simpler rule of thumb based on the ratio of the planet's escape velocity divided by the of the particles of a gas in the exosphere of the atmosphere, the layer that gas particles escape into from. The root-mean-square velocity of gas particles in the exosphere depends on their temperature in the exosphere, which seems to be higher than the surface temperature of a world.
According to table 5 on page 35, a comparatively minor change the ratio can make a comparatively big change in the length of time it takes for the amount of gas to be reduced to 1/e, or 0.368 of its original amount.
According to table 5. if the ratio is 1 or 2 the time to reduce the gas to 0.368 of hte original amount is zero, if the ratio is three a few weeks, if the ratio is four a few thousand years, if the ratio is five about 100 million years, if the ratio is six infinite time.
If an atmospheric gas is lost slowly enough, it might be replaced from various sources as fast or faster than it is lost. And there are various pocesses which can speed up atmospheric loss.
So for each kilometer per second of escape velocity a world has, the mximum root-mean-square velocity of gas particles in its exosphere should be no more than 0.1666 to 0.2 kilometers per second, if the world is going to retain that gas for signifiantly long time periodss.
On pages 53 to 58 Dole discusses the mass range for a human habitable planet. He decides that humans wouldn't settle a planet with a surface gravity higher than 1.5 g, 1.5 times the surface gravit of Earth. Dole calculated a planet with a surface gravity of 1.5 g would have 2.35 the Mass of Earth, 1.25 the radius of Earth, and an escape velocity of 15.03 kilometers per second (which is 1.367 times Earth escape veloity of 11.186 kilometers per second).
The average surfce temperaure of Earth is about 288 degrees K (Kelvin). on page 54 Dole says the expshere temperatures of Earth are bout 1000 K to 2000K. Dole says that if a planet can have an Earth like surface temperature of about 288 K and a maximum exopshere stemperature of 1000k, the root-mean-square velocity of oxygen atoms in the exosphere would be about 1.25 kilometers per second. A world would need to have an escape velocity five times that, or 6.25 kilometers per second, to retain 0.368 of its original oxygen atmosphere for about 100 million years.
So Dole selects an escape velocity of 6.25 kilometers per second (0.558 tha tof Earth) that of Earth) as the minimum for a human habitable world. According to Dole, that corresponds to a world with 0.195 the mass of Earth, 0.63 the radius of Earth, and a surface gravity of 0.49 g.
Dole didn't believe that a world that small could produce a dense oxygen atmosphere. Two different calculations on page 56 gave lower mass limits of 0.25 and 0.57 Earth mass to produce a dense oxygen rich atmosphere, and Dole rather arbitaraily selected 0.4 the mass of Earth as the minimum mass to produce an oxygen atmosphere, correponding to 0.78 Earth radius and a surfce gravity of 0.68 g.
More recent discussions of planetary habitiablity discuss habitability of liquid water using life in general. Any sort of atmosphere will do for such a planet, as long as it is dense enough to support liquid water on the surface and has the right influence on planetary temperature for water to be liquid on the surface. So an atmosphere with a lot of oxygen is not a requiement for a world to be habitable in such discussions.
This article from 2013 https://arxiv.org/ftp/arxiv/papers/1209/1209.5323.pdf has a paragraph discussing the mass range for habitable worlds on pages 3 to 4:
A minimum mass of an exomoon is required to drive a magnetic shield on a billion-year timescale (Ms ≳ 0.1M⊕,
Tachinami et al. 2011); to sustain a substantial, long-lived atmosphere (Ms ≳ 0.12M⊕, Williams et al. 1997; Kaltenegger
2000); and to drive tectonic activity (Ms ≳ 0.23M⊕, Williams et al. 1997), which is necessary to maintain plate tectonics and
to support the carbon-silicate cycle. Weak internal dynamos have been detected in Mercury and Ganymede (Kivelson et al.
1996; Gurnett et al. 1996), suggesting that satellite masses > 0.25M⊕ will be adequate for considerations of exomoon
habitability. This lower limit, however, is not a fixed number. Further sources of energy – such as radiogenic and tidal heating, and the effect of a moon’s composition and structure – can alter our limit in either direction. An upper mass limit is
given by the fact that increasing mass leads to high pressures in the moon’s interior, which will increase the mantle viscosity
and depress heat transfer throughout the mantle as well as in the core. Above a critical mass, the dynamo is strongly
suppressed and becomes too weak to generate a magnetic field or sustain plate tectonics. This maximum mass can be placed
around 2M⊕ (Gaidos et al. 2010; Noack & Breuer 2011; Stamenković et al. 2011). Summing up these conditions, we expect
approximately Earth-mass moons to be habitable, and these objects could be detectable with the newly started Hunt for
Exomoons with Kepler (HEK) project (Kipping et al. 2012).
And the articles cited should give reasons for those mass limits.
I note that Titan, the largest moon of Saturn, has a mass of only 0.022t that of Earth, a surface gravity of 0.138 g, and an escape velocity of 2.641 kilometers per second, 0.236 times that of Earth. And it has dense atmopshere, a little denser than Earth's, and billions of times more dense than the atmospheres of hte similar satellites Ganymede and Callisto.
Of course Titan gets only about 0.01 as much energy from the Sun as Earth does, and its surface temperature is 93.7 K, much colder than Earth, which means that its exopshere temperatures, and gas velocities, should also be much lower than Earth's.
And if there was a way for a world with the escpae veloicity of Titan to have EArthlike surface temperatures and also exosphere temperatures a low as those of Titan, it would be able to retain an oxygen atmopshere for geological eras of time. But I don not know of any way to make that possible.
In 2019, a new study suggested a new lower mass limit for habitable worlds. But it applied only to water worlds, with world wide oceans many kilometers deep covering their solid surfaces. Such worlds could be habitable down to a lower mass limit of 0.0268 Earth mass.
And that is smaller than your 0.13 Earth mass smaller world.
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