Might it be possible for ShareWith to authenticate with my application using some simple method (e.g. a shared secret HTTP header)? Then I could get all the benefits of the auth model without having to install anything (other than my app).
In theory, you could provide an auth header representing your incoming user to our API and verify that way, yes. You'd still need to perform authentication (using Auth0 right now, as that is what ShareWith uses) to identify which user is hitting your application, but then the authorization check could be done via the API.