Hey Steve,
Yes, mechanically everything you have mentioned is correct, players roll initiative, then act in sequence just like in combat. Here are my suggestions for this particular encounter:
Rather than having a "tug of war" track, have 1 long one. Put two tokens at the "start" space, 1 to represent the strength of the PCs arguments, the other to represent the nobleman's "patience". Each time the PCs "influence" their target (the nobleman) move their token up a space (maybe an additional space if it was a great success or they got a Sigmar's Comet). If the PCs get this token to the end of the track, they've "won" the encounter and the baron spills what he knows (or something similar).
At the end of each round have the baron's patience move up 1 space, possibly also move it up if the PCs get Chaos Stars or fail with lots of Banes. Sigmar's Comets might also make the token move backward or "pause" for a round. When patience marker reaches a certain number of spaces down the track (say 5 or 6), he finally loses his patience with the PCs and demands they leave ("My butler will see you out, NOW." sort of thing), thus ending the encounter in "failure".
On the baron's turn, he could instead of trying to influence either tracker, begin attacking the PCs personally (making them rack up Shame). If a PC goes over their Shame Threshold, they're out of this encounter (either they leave the room in a huff or the baron no longer cares one whit about anything they have to say). The baron could also try to derail the PCs arguments with counter points that could move their progress tracker backwards. He could also begin to try and "escort them out" or cow them into leaving quicker, moving his patience tracker upward more quickly.
This way it isn't 3 vs. 1 in a tug of war. Instead the PCs basically have a limited amount of time (represented by the baron's patience) to influence him enough to get what they want. He still has an impact on the encounter, social PCs get to shine, and it can be a tight race to the finish to see if the PCs can convince him to give them the info before he throws them out. If "throwing them out" isn't practical, simply refusing to talk to them anymore could also work.
How far to make each event space down the track depends on how successful you think your PCs will be at influencing him and how hard you want the encounter to be. Figuring that each PC will move the success token once per round is probably a good start (so 3 times a round). Some might fail to move it naturally, but spectacular successes or Sigmar's Comets might move it more than once to. I would also have the first event space (the one that if the baron's patience token reaches the encounter ends in "failure") do something for the PCs success token when they cross it. Maybe they are wearing the baron down and get white dice to all social actions for the rest of the encounter, maybe their arguments seem to be working and his patience goes backward a space, etc.
You could also have two separate trackers instead, one for each token. Then just make the PCs 3 times as long as the barons, thus they have to be 3 times as successful to "win" the encounter.