[psycopg3] Test that the SDK does not cause an error on DB reconnect

We should add a test for the changes implemented in #3111. We are not adding the test in that PR, since testing this change is somewhat difficult, and it makes little sense to allow the lack of a test block the release of the bug fix.

I have added my in-progress work on the unit test in #3138 – this draft should be used as a starting point