Extra Hitter

[bklynump]

The league I work uses an extra hitter. It is optional and neither team is required to use it. During a game this season, Team A opted to use the EH, and batted 10 players. However, the lineup they submitted to the umpires and Team B, only listed 9; the EH was not included. Team B did not notice this until Team A had batted once through the order with 10 players. The second time around, when the EH cam up to bat, Team B argued that Team A was batting out of order.

My question is, how would you handle this situation? There are a couple of options:
(1.) Once Team A batted through the order once, the EH was legalized, as if he were an unannounced subsititute.
(2.) Handle it like a batting out of order situation. Removed the EH from the game. Call out the proper batter, and the player who was the EH is now on the bench.
What would you do?

Thanks.