Since I don't have time to actually code this idea right now to make it play and see how it works, a slightly longer description of what I had in mind: not so much dancing as mingling, with the intent of trying to gain social influence with all sorts of factions.
So, you're in a big reception room / ballroom. There are many guests. You want to get to the important people to schmooze with them but it's crowded and there are many people in the way, so you'll have to encounter and defeat other folks as you run into them. There are many approaches you can use to try and defeat someone, from flirting to impressing them with your knowledge to 'accidentally' bumping into them and denouncing them for it, and a lot of this would depend on skillchecks.
If you lose the battle you have to withdraw from the ballroom for the week. If you win then your opponent gets out of your way, adjusts your influence with their faction (like XP - important people will give you a lot more influence for beating them) and, if you're lucky, gives you a present. (LOOT!) And then you can move on to the next fight.
This would therefore allow for all the trappings of turn-based RPG combat but in a social situation rather than a cutting-off-heads situation, since a ruling-focused LLTQ-type game can't really have the heir wandering off into dungeons. :)