If Azure is rewriting your machineKey, you can't do much about it, as it is part of their infrastructure. However, there are other methods.
Override FormsAuthentication
This should not be difficult as you can easily look up for source code of FormsAuthentication and create your own logic and replace MachineKey with your own key stored in web.config or in your database.
Custom Authentication Filter
The simplest way would be to create a filter and check, verify, encrypt decrypt cookies in your filter. You need to do this on OnAuthorization method and create new instance of IPrincipal and set IsAuthenticated to true if descryption was successful.
OAuth
- Enable OAuth and create OAuthProvider. However you will need to host OAuthProvider on server that is in your control as that will need machineKey working.
- Enable Third Party OAuth, if you enable OAuth with Google, Facebook etc, it will be easy as user will be redirected to OAuth provider and they will continue to login automatically and a new session will be established.