Here's my question. Why should they trust the new character? I find far too often players ignore the role playing aspect (we all been guilty of it honestly at times) and assume they should. Some of the best role playing though can come out of "earning" the trust, or even just using the mistrust as part of the game. As the GM, I'd just go with it and try to work it into the story. Give them reasons to work together or not as they see fit, and however they go with it, make that work within the story.
Personally, a Player vs player fight can be very rewarding anyways
Just decide ahead of time if they get to actually kill each other, or if you will miraculously save a character unconscious. Sometimes someone may not enjoy their character and choose to reroll, and that can be a fun way to end it.