Nope, the rotation of the central body typically has negligible effects on its surroundings (on a short time scale).
Technically you're right, assuming the exit-point is stationary in ecliptical coordinates the Indus would be put on an orbit around the Sun depending on its emergence vector. The Sun's rotational period at equator is 24.47 days, so most objects in the Solar system orbit the Sun much slower than the Sun itself rotates. However, prograde or retrograde orbits in this case has little meaning since Sun's surface is fluid rather than a solid reference point.
Actually, you can get absurd effects when you have a rotating supermassive black hole, because all rotating objects twist the space-time around them to some extent, but with normal celestial objects and normal astronomical distances, these effects typically don't appear.
And yeah, tidal forces can cause all sorts of complex effects. If we consider the Earth-Moon pair, one object (Moon) is tidally locked (which would probably not be the case with Indus) on prograde orbit. In this case, the tidal forces do work to slow down Earth's rotation, so Earth's rotational energy and angular momentum are decreased.
However, both the mechanical energy and angular momentum of the Earth-Moon system conserve; the conservation of angular momentum actually means that while the Earth's angular momentum decreases, the Moon's angular momentum increases, which means the Moon is being pushed to higher orbit. As far as energy is concerned, most of the reduced rotational energy is converted to heat, but about 1/30th of it is converted to the Moon's potential energy as it slowly climbs to higher orbit.
Tidal forces on the Indus would probably not be observable in the time scale of Sunglare; the actual problem they are having must be something more acute.
Personally I would put a wager on two possible (and one so-so) causes:
1. Indus is simple descending too low on elliptical orbit
2. Indus is venting something to retrograde direction, causing their orbit to decay.
3. Indus' magic antigravity systems are damaged and thus it has become subject to Newton's laws of motion as well as the universal gravitational law, unlike in all other appearances.
We've seen stations hold stationary altitude at orbits that are clearly not geostationary... the ships are firing their thrusters all the time yet experience no constant acceleration... a "decaying orbit" in this environment can mean whatever is required to make the situation as dire as possible.
