Mach's Principle and spin angular momentum

Your question is written as if Mach's principle were true, but in fact it's false, in the sense that general relativity doesn't obey Mach's principle, and GR is verified by experiment, while more machian theories like Brans-Dicke gravity are falsified by experiment.

So if we were to answer your question, we would have to (1) come up with some new theory of gravity that works at least as well as GR and is more machian than GR; (2) quantize that theory.

If you want to know how intrinsic spin transforms when you go into a rotating frame, then it's not a big mystery. This is a common thing to do in nuclear physics. We add a term $-omega J$ to the Hamiltonian, where the angular frequency acts like a Lagrange multiplier that can be used to control $langle J rangle$. This new Hamiltonian has different eigenstates with different structure, but spin $-1/2$ particles are still spin $1/2$.