I'm in a similar situation with a project I'm working on where I need to authenticate players. I'm not sure what the 'official' Itch view is.
As a third party website inside an iframe, you face restrictions to some web APIs/functionality like third party cookies in some browsers, which won't be an issue if you direct your users to your website at the top level. Instead, you could have a web game upload where your game page just shows a link to your own website that the user has to click.