Okta expressions: changing username passed to external IDP


I am doing inbound federation with an external IDP, and I want the okta usernames to be of the format domain\email - for example GMAIL\gopa@gmail.com - ie, the user will try to login to okta as GMAIL\gopa@gmail.com

BUT - when the username is sent to google IDP, I want it to be just gopa@gmail.com - ie I want the GMAIL\ removed. I do know about profile mappings and regular expressions there - but profile mapping is when something is about to be “saved” in okta right ? This is something I want to do before anything is saved anywhere - ie at the time of sending the username to the IDP ? Can this be done ? Just to be clear- I need the username STORED in okta as GMAIL\gopa@gmail.com - but SENT to IDP as gopa@gmail.com. Any links/references will be useful please