Thank you for taking the time to give such a complete answer.
The login is an optional feature intended to reward the user with additional and personalised functionality in return for registering. Thus the login is not a gateway, but a feature on the page to unlock additional content.
A user might be in the middle of reading content when they need to login/register to interact with user specific content. It would be OK, if not ideal, for the page to reload, but closing the login would have to return the user to the same page and we would very much prefer for the login to be overlayed on the page. So this is not so much about revealing the Auth0 domain as it is about providing a seamless user experience.
Thank you for the embedded authentication link. I suspect this is the route we will have to go.
Because our website is fairly large and uses webpack one issue we came across when we installed the auth0-js package with npm our newer web-pack v3.0 did not work with your pre-built version which seems to use web-pack 1.0. The global object injected by web-pack did not get injected, so we had to import from the source so our newer web-pack version could bundle it, like so:
import WebAuth from "../../../grunt/v2/node_modules/auth0-js/src/web-auth/index"
Also, when I tested installing using npm auth0-lock the auth0-lock node-module did not appear to be built and I think this is the module I would need for building an embedded lock.