In my experience, reverse proxying Plex is far more trouble than the benefits are worth. Honestly, I just use app.plex.com.
It looks like they’ve written their web app to issue redirects to their own site. You could try to use proxy header regex rewriting (see: https://github.com/mholt/caddy/pull/2144), which should be available in the latest Caddy release (although currently still undocumented), to alter the Location header it sends and force it to redirect to your own site. No promises that the locally hosted version served by the Plex instance itself even has a login path, though.