Multiple bugs in `multiprocessing.managers`: pickling, nesting, etc

Bug report

Bug description:

I noticed multiple bugs in multiprocessing.mangers and explained them, with fixes, in a blog post: https://zpz.github.io/blog/python-mp-manager-2/

A summary of the bugs are as follows:

• Pickling of BaseProxy objects: easily causes object release before unpickling happens in another process
• Unable to call proxy-returning methods on a proxy that has been passed to another process. (There is an open PR https://github.com/python/cpython/pull/99598 but my solution appears simpler and cleaner.)
• "Nested proxies" are never released (also reported in this isseu https://github.com/python/cpython/issues/103136)
• A simple enhancement (not exactly a bug): server-side exceptions lose all traceback info when propagated to user (also reported here in this issue https://github.com/python/cpython/issues/78279)

In addition, this issue

• The specification of exposed is a mess. (See https://zpz.github.io/blog/python-mp-manager-2/#exposed-methods)

My package mpservice.multiprocessing.server_process fixes these issues with some additional enhancements and simplifications. The enhancements are covered in another blog post: https://zpz.github.io/blog/python-mp-manager-3/

I hope to engage in some discussions (esp with a core developer) before working on some PR, either local fixes or something a little more holistic.

CPython versions tested on:

3.10, 3.13

Operating systems tested on:

Linux

Start with the multiprocessing.managers entry points and the linked blog post, then compare the reported behaviors with the referenced PRs and issues. Reproduce the pickling, nested-proxy, exception-traceback, and exposed-method cases; the work is done only after the scope is agreed and each selected bug has regression coverage.