When I am in GM only mode
 

If I am only acting as the GM, and I am experiencing a rash of pitcher injuries, or need to get my rotation back in order. Is there a way I can tell my manager to have a “bullpen day”.

Or would I have to temporarily take control of the team and put a long reliever or middle reliever into a starting pitcher role?

If you have legacy mode GM controls lineups enabled you can insert a RP into the rotation as the next SP.