Second invariant rate of strain type material coefficients are a pain. See for example Default absolute tolerance and relative tolerance - #4 by nate. Your problem is highly nonlinear and I expect will be quite challenging to get a Newton solver to converge without “encouragement”.
If I were you, I’d try using the solution from “choice 2” as the initial guess for “choice 1” along with a damping parameter, and see if that gets you any closer.