Betting is the replacement for dueling.
It’s not a perfect replacement, of course. (Nothing is a perfect replacement for anything else.) It only applies in certain cases. But what it has in common with dueling is the challenge either to back up one’s opinion or retract it. In that sense, it serves a similar social function, and gives the challenger a similar feeling of satisfaction. And in addition to being (obviously) morally preferable to dueling, a challenge to wager also makes more sense epistemically. When a challenge is accepted, the outcome of the wager can show who’s right, whereas the outcome of a fight doesn’t (unless the wager is about relative fighting prowess, but in that case the duel just is a wager). And when a challenge is refused, well, fear of being refuted is an epistemically relevant reason to retract an opinion, while fear of being killed is not.
I was thinking I might be able to add something insightful from my knowledge of poker theory. However, it seems that there’s actually very little overlap at all because poker always has multiple rounds of betting; sometimes only two, and generally no more than 5. But the wagering you’re talking about is generally one-off, and therefore aspects of bluffing that are so essential in poker (though still way overestimated in importance by most players) become almost completely irrelevant. The only form bluffing takes in a one-off wager would be to convince the other person not to wager with you at all; basically, you’re bluffing in the hope of not losing anything.
In that sense, perhaps Pascal’s Wager should be re-named Pascal’s Bluff.