You can use a frame that translates on a circle with the ship, but doesn't rotate. This will produce a uniform gravitational field with a time dependent direction. However since it's time dependent, you cannot assign a potential to it. You can just argue that the asteroid is always higher in the field.Using an apparent uniform (but constantly changing due to the "orbit") gravitational field with attendant time dilation effects is one way to solve the problem, but obviously not the only way.
The co-rotating common rest frame has a non-uniform but time independent gravitational field, so you can assign a potential it.