Rebinding a SQL Server user to its login after a restore with this user
Unfortunately no.  Remember that a database principal and a server principal are two separate objects.  Until you fix the orphan the server principal MyUser has no access to MyDatabase even though there is an orphaned user called MyUser in the database.  

The other way to fix an orphan is to change the SID of the server principal to match that of the database principal.  This does have a risk of creating orphans in other databases and even more importantly in this case to do it you have to drop the principal and re-create it with the new SID.  Which of course isn't possible using that particular ID.

Your only method is going to be to fix this using a separate id with the appropriate permissions.  That could be an id with sysadmin permissions (which would have permissions to connect to the database even without an associated database principal) or an id that has DBO access on the database.  Note those are examples, there are several other permissions available that will allow you to modify server/database principals.