Since the group is small and the players are happy with the IC relationship, do not reconcile the characters. Simply give them reason to be in the same place at the same time and let them oppose each other or form a temporary truce as they please. Perhaps eventually they will genuinely reconcile, but I don't think it's really for the GM to decide that's what the plot of the session is, since if it happens it has to happen in a way that's realistic to each player.
should I introduce a dire threat that forces them together despite all of their differences and dislike
That's reasonable. Or perhaps they all want the same thing and co-operate against other parties to get it, knowing that eventually they'll break truce and compete (e.g. The Maltese Falcon). Or they overtly compete, but politics/diplomacy/law dictates that they be somewhat civilized about it (e.g. The Lion in Winter / Empire).
how should I introduce it? During the session, or before it starts as the starting point for it?
Just for the sake of character POV, I'd always prefer to introduce the news of the threat in session, with the camera rolling, so that players can react in character. I would not say, "between sessions you've all reconciled, forgiven each other for the events of last session, and become great friends again", because even if the players buy into the goal of starting afresh, it won't be very real to them.
In this example the characters presumably will be apart until brought together, so that means two (or three) short introductory scenes before they come back together. If your twisty turny plot requires information to be given secretly to the players then you might want to do those before the session to save session time, otherwise keep it in the open and let the players enjoy the other side's POV even though their characters aren't present. Give each character a reason to be in the place the plot is going to happen, then bring them together. If it's going to take so long to bring them together, that you think it'll be too much time spectating in-session, then certainly handle it outside the session if you can. My limit's about one scene each, two at a push. Unless you're running Ars Magica, you don't want to be switching between threads that stay separate for ages. The point of most RPGs is to cover a group of people who are actually interacting!
If the group was not small (so that managing lots of PCs all running in different directions was logistically difficult), or the players were unhappy (and would prefer to play a united party), then I think some kind of GM relationship counselling might be a good idea, but here it seems more interesting to stick with the source of drama that's working. As in any game, watch for signs of the players becoming unhappy. I'd be especially concerned if the "score" reaches 3-0 to the Monk, there's a risk that the other two don't mind being beaten a couple of times, but won't want to indefinitely play a game in which their role is to lose. Partial victories for both sides are probably more stable, but it depends what the players like about the game.