My pawn is pinned by the enemy bishop, however if I push the pinned pawn 1 space my rook would then put the opposing king in check resulting in a simultaneous check with my opponent to move next Can I move my pinned pawn in that situation?
No. Doing so would put you in check, and it would be the enemy's turn.
You can think of it this way, the enemy would have the first opportunity to take a king, so if the game was about taking kings, then you would lose in this situation.
You can never make a move that would allow the enemy to take your king, even if you put their king in check at the same time.