Facing the Earth and facing the sun are highly correlated. Seeing one xor the other is possible, but unusual.
I expect that window is small to the point where no one actually cares. It would be an interesting applied-math problem for a grade-schooler, though.
The angular separation between the Sun and the Earth as seen from a space probe is significant as far out as Cassini at Saturn. The probe can receive commands from Earth without the signal being overwhelmed by solar radiation, except for a few days each (Earth) year when Earth is too close to the Sun as seen from the spacecraft. (Earth doesn't literally go behind the sun often, thanks to inclination of the planetary orbits plus Cassini's own inclined orbit at Saturn.)
> Our line of sight to Mars is independent from Mars seeing the sun.
Our line of sight to Mars as an entire body is independent, but our line of sight to a particular point on Mars is indeed correlated with that point facing the Sun.