When playing matching in local multiplayer, if second player loses they can press a button on their arcade stick / controller to rejoin the match. Now the issue is that when the first player loses the match, they cannot rejoin local multiplayer. We have the same control setup, but he can't join again. Is there any way to fix this?

No need to worry about this post anymore. The OP and I got it fixed. Apparently, player 2 has a different button used to rejoin the match (in our case).