Link to related feature request: Show login form in popup window
Are you familiar with the Custom Signup library? It’s an app you can clone that allows you to create your own login forms and other user flow elements. You might be able to use it to create a login form that looks good in your iframe.
https://anvil.works/ide#clone:TTLCYBO7UWVCSTWL=T37TCLU5COZYUZ2WONZ7RKXU