[3.13] gh-122201: Lock mutex when setting handling_thread to NULL (GH-122204) (#122319)

In the free-threaded build, we need to lock pending->mutex when clearing
the handling_thread in order not to race with a concurrent
make_pending_calls in the same interpreter.
(cherry picked from commit c557ae97d6bd9d04164a19b4fe136610e54dbdd8)

Co-authored-by: Sam Gross <colesbury@gmail.com>