Improve MultiIndex label rename checks
- [x] closes #55169 
- [x] [Tests added and passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#writing-tests) if fixing a bug or adding a new feature
- [x] All [code checks passed](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#pre-commit).
- [x] Added [type annotations](https://pandas.pydata.org/pandas-docs/dev/development/contributing_codebase.html#type-hints) to new arguments/methods/functions.
- [x] Added an entry in the latest `doc/source/whatsnew/vX.X.X.rst` file if fixing a bug or adding a new feature.
@mroeschke - this is ready for your review This pull request is stale because it has been open for thirty days with no activity. Please [update](https://pandas.pydata.org/pandas-docs/stable/development/contributing.html#updating-your-pull-request) and respond to this comment if you're still interested in working on this. Thanks for the pull request, but it appears to have gone stale. If interested in continuing, please merge in the main branch, address any review comments and/or failing tests, and we can reopen. Hi @mroeschke,
The merge conflict was only against the `doc/source/whatsnew/v3.0.0.rst` file which I've fixed.
After the merge, pandas is failing to compile on my dev environment due to what I'm assuming is an unrelated error:
```
[9/71] Compiling C object pandas/_libs/interval.cpython-311-x86_64-linux-gnu.so.p/meson-generated_pandas__libs_interval.pyx.c.o ninja: build stopped: subcommand failed.
```
I'll rebuild my dev environment and update once I can confirm that tests are still passing