Refactor the code to use RecursiveMutextGuard to make it more readable
and less error prone.

This is a tentative fix for a rare deadlock that appears in
test-cpu-profiler/CrossScriptInliningCallerLineNumbers.

Bug: v8:11191
Change-Id: Ia32e7f61167f95e0fce142992c83ddff11959222
Reviewed-on: https://chromium-review.googlesource.com/c/v8/v8/+/3779690
Commit-Queue: Camillo Bruni <cbruni@chromium.org>
Reviewed-by: Patrick Thier <pthier@chromium.org>
Cr-Commit-Position: refs/heads/main@{#81991}