The nature of space-time is apparently such that it need not necessarily follow the ordinary laws of physics. ie, galaxies can and do expand away from each other faster than light speed. This is so because the galaxies are not actually moving through space but rather that the space between them is growing larger. The 'dimensionality', or 'measuring stick', is increasing in magnitude.
Well, what you talk about doesn’t actually violate relativity, since you are dealing with different reference frames, such as the old thought experiment with two ships leaving earth in opposite directions, each traveling 0.9c. If you imagine a stationary observer with a universal reference frame, it seems to violate relativity, but there is no universal reference frame in relativity, so it’s just an illusion of a violation.
On the other hand, acceleration requiring a force isn’t a principle that varies with reference frames or depends on relativity. It’s simply basic mechanics. Something is accelerating the matter in the universe, and it doesn’t matter how that acceleration is transmitted, through the expansion of space or otherwise, it still requires a force.