That's a good question. For this, you should not use the supertokens-auth-react SDK. Instead, use the supertokens-website SDK which doesn't require a websiteDomain, and only requires an apiDomain.
You can follow the "Plain Javascript" code snippets in the docs for that.
Finally, i'm not sure that it will actually work with a browser extension. We have never tried it with that. But in case it doesn't I'm sure that there is some solution we can figure out. Lmk