Three of the necessary factors for a roleplaying group to work are:
- That the players can get along enough to play the game
- That the characters can cooperate enough to achieve party goals
- That the way the characters act and the story progresses is credible and consistent enough that players can suspend disbelief
Your DM is expressing to you that character cooperation is a problem, and is worried that this will lead to player tension or just people not enjoying themselves. You are worried about consistency, which is important - but you still need to satisfy the other concerns. It might be that everyone round the table is 100% fine with the tension between the characters, If you're sure that's true, fine. But it sounds like that might not be the case.
So talk to the DM and the other player, out of character and outside of the session. Start by establishing trust: agree that you all have an interest in finding a way for your characters to cooperate that fits the narrative. Talk about the reasons why they currently don't, the problems that causes, and what could change that might make them cooperate better.
Ideas for events that might trigger the change:
- The characters realise they are going to be working together for a while and gradually open up.
- The characters are forced to exchange revelations about themselves which lead to them respecting each other more
- One character makes a serious blunder which endangers the party. They apologise, and look for an opportunity to make amends.
- One character blames another for endangering the party. They all have a massive argument... and eventually everyone regrets the tensions and sheepishly apologises.
RP is not just about expressing your character concept in its pure, immutable form, but about telling how your character learns, grows and changes in response to new and unexpected experiences. If the thought crosses your mind that "there's only one way my character can react" take that as a challenge to find a way to confound expectations. Try to find something all three of you can agree on, and if something seems wrong, talk about why.
You should try hard to reach a clear conclusion, probably one which fits neatly into one of these categories:
- It goes well and the DM is confident they can trigger a situation that will resolve the conflict and fits with the plot (and you'll have to wait to find out what it is!).
- It goes well and you between you come up with an idea that will resolve the conflict. You perhaps don't know exactly how it pans out in advance but you both agree as players that the end result will be that the characters' respective 'antisocial behaviours' are dropped.
- You decide that the conflict is not solvable. The characters cannot possibly work together well enough for you to enjoy playing the game and you need to decide how to fix the enjoyment issue. One (or preferably both) of you plans for your character to leave the group.
Remember, you can always tear up a character sheet and roll up a new one. Not so for your real-life relationships.