If, during recruiting, a player ends up drawing two action cards of characters that are already in play, is the player stuck with losing a character for the remaining of the game?

No, you just continue to draw cards, one at the time, until you get a leader (check "Recruiting" section of the Rules Reference).

I leave you the refference from the RRG, page 11 for what you can do when you already recruited the leaders from both cards:

A player cannot recruit a leader that is already on the game board, in a leader pool, or eliminated from the game.
• If all of the leaders shown on the player’s drawn action cards have already been recruited, the player may continue drawing action cards one at a time until he draws a card that shows a leader that he has not already recruited.
• A player can choose an action card even if he cannot recruit any of the leaders on the card. He gains the card but does not recruit a new leader.