“claim[ing] that special relativity can't handle acceleration at all ... is like saying that Cartesian coordinates can't handle circles”

See http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html

But then again, the question whether the study of flat spacetime using non-inertial reference frames counts as SR depends on what you mean by SR. If you mean the limit of GR as G approaches 0, then it totally does.