In addition to the Nicolas’ suggestions, maybe you can try using direct solver, just to see if the problem is not caused by iterative solver (although, based on your code, it is possible that you have already tried this before).
Alternatively, you can try using PETScSNESSolver(), which has a bit more options than built-in NewtonSolver, or you can designing your own NewtonSolver as proposed here.
Regarding the solver parameters, you can pass them alternatively in the following way (example is for PETScSNESSolver()):
nonlinear_solver = PETScSNESSolver()
PETScOptions.set("snes_rtol ", "1e-4")
PETScOptions.set("snes_atol ", "1e-4")
PETScOptions.set("snes_monitor")
PETScOptions.set("ksp_initial_guess_nonzero", "true")
PETScOptions.set("ksp_type", "gmres")
PETScOptions.set("pc_type", "hypre")
PETScOptions.set("ksp_monitor")
PETScOptions.set("snes_converged_reason")
nonlinear_solver.set_from_options()
Beware that for some reason setting the relative and absolute tolerance for nonlinear solver in this way does not work, so you will need to do it as in your example. Hope this helps.