Right, I figured out the issue.
You cannot use that connectionstring for anything else than the edmx connection. The way I have it in my web.config makes the website use that same connectionstring for the membership stuff too, which is not compatible with the System.Data.EntityClient provider and needs the System.Data.Sqlclient in my case. Adding a second connectionstring, minus the Entity Framework stuff in it and having that referenced by the membership provider attributes in the web.config removes all errors and allows the page to render and request stuff from the SQL server.
Nightmare over, can return to work on my website. o/