Instead of finding one approximation of arcsin over the whole range [0,1], consider subdividing the range and then finding a separate approximation over each subrange. This is how splines work. Ensuring monotonicity where the separate approximations join can be nontrivial.
In any case, since we've already approximated arcsin over [0, 0.7071], let that be one range, and let the other range be [0.7071,1.0]. Let's try to find all minimax rational approximation with total degrees from 1 up to xxxx.
These are the only cases, up to total degree 11, were Maple succeeded with default Digits: [0,1], [1,0], [0,2], [1,1], [2,0], [0,3], [3,0], [0,4], [4,0], [0,5], [0,6], [6,0]