The only sense in which special relativity is an approximation when there are accelerating bodies is that gravitational effects such as generation of gravitational waves are being ignored. But of course there are larger gravitational effects being neglected even when massive bodies are not accelerating and they are small for many applications so this is not strictly relevant. Special relativity gives a completely self-consistent description of the mechanics of accelerating bodies neglecting gravitation, just as Newtonian mechanics did. The difference between general and special relativity is that in the general theory all frames of reference including spinning and accelerating frames are treated on an equal footing. In special relativity accelerating frames are different from inertial frames.
Velocities are relative but acceleration is treated as absolute.
In general relativity all motion is relative. To accommodate this change general relativity has to use curved space-time. In special relativity space-time is always flat. In special relativity an accelerating particle has a worldline which is not straight. This is not difficult to handle.