Solved
This turned out to be easier than I expected, oops. I modified signin_inner.tmpl to only have the small OAuth2 portion of the signin page, i.e. I deleted lines 13-54 and 56-58 from that file (as it exists now), and put my version in gitea's folder custom/templates/user/auth/. As desired, it still displays "Login with cloudron", but not the main login option.