Have you tried to see if using direnv.el instead of the envrc package fixes the lsp-bridge issues? If it does, it's probably an issue related to how envrc interacts with temp buffers. From the envrc-mode README:

...it's possible you've found code that runs a process in a temp buffer and neglects to propagate your environment to that buffer before doing so.

A couple of common Emacs commands that suffer from this defect are also patched directly via advice in envrc.el — shell-command-to-string is a prominent example!

The inheritenv package was designed to handle this case in general.

I haven't tried any of this myself, so this is just speculation.