[zlynx]

In relativity, everything is relative. You can choose any "rest frame" that you like and all the math still works out.

For the star, from its point of view its mass has not changed at all. So yes fusion still works.

[jniedrauer]

From the perspective of the star, what impact does relativistic speed have on the rest of the universe? Does it also gain mass? Does that mean that an object accelerating to relativistic speed increases the mass of the universe from both frames of reference?