Refactor `test_pydantic_ai` to remove usage of internal functions within tests

[ericapisani]

Related: https://github.com/getsentry/sentry-python/pull/5629/changes#r2917105295

There's a couple of spots that are using the internal function _set_input_messages that need to be cleaned up.

Ensure that all tests are using public APIs/functions

[linear-code]

PY-2132