Yes, backward Euler could work. It all depends on the stability of your scheme. You could also use the Newton solver, or implement a fixed-point iteration.
In the forum you could find some post on non-linear time-dependent problems:
https://fenicsproject.discourse.group/t/nonlinear-heat-equation-performance-dolfin-vs-dolfinx/11961
https://fenicsproject.discourse.group/t/how-to-check-if-the-nonlinearproblem-is-setup-properly/10218
Also, in this dolfinx-tutorial by @bleyerj you can find a thermo-elastic evolution problem.
Cheers.