Short answer: no, because physics says so.
Explanation: Each celestial object can be looked at as a system (an expression) with it's own properties: mass, speed, rotation speed etc. Earth has those properties, Solar system has those as well (only the numbers are BIGGER). Let's assume a fighter is a system also (it has same properties, after all). Fighter must adjust it's parameters to the bigger system - Death Star in this case - which is done by entering it's gravitational field first, atmosphere second (in other words: it will slow down and heat up from atmospheric friction and will be bound by gravity). At this time it becomes a part of the larger system and enters relational relation. It will move only in relation to the Death Star's surface, but from outside they both will travel through space at same speed, rotation etc.
Fighter as part of the system is subject to changes with those just like everything else. So if Death Star would start to rotate faster, fighter will also, as it's moving through the atmosphere but ALSO WITH the atmosphere.
Obviously there will be stronger Coriolis effect, affecting the fighter, but on the whole... It wouldn't be a problem, even increasing rotation speed by order of magnitude (or more).
And yes - Death Star entering the solar system (any) would have to adjust to it as well. However, this is a bit different effect than simple atmospheric braking... It's called frame dragging and it's really cool, especially around heavier stars. But never mind that. Just a fun fact.
Never mind also the fact that rotation speed of a celestial body affects it's shape... Yes, Earth is not round. It's a geoid, flatter on the poles and wider at the equator. That is why gravity is not uniform across the globe's surface. Had the Death Star started rotating much faster than earlier it would become more flat as well. Won't even speculate what would happen to the thrench then... Another fun fact only.
As a side note: increasing rotation speed would also decrease gravitational forces (by increasing upward acceleration), so fighters would be faster and more nimble, too... Yet another fun fact.