Courts interpret the law, they don't enact or enforce it.
This sounds much more like a town-hall meeting, which will probably enforce some variant on Robert's Rules of Order. To a certain degree, trying to shift things once the meeting has begun will be a waste of time -- many of the key decision makers will already have their positions locked in place. So make a virtue out of that!
I'd probably give the player a set time frame before the meeting, then let them use that time to try and manipulate events ahead of time. A series of social 'conflicts' with city counselors, lawyers, citizenry -- let them be creative. Depending on how well / poorly they do at each location, they eat up different amounts of time at each location. If they do exceptionally well, maybe it takes only minutes to convince their new ally to join. If they stubbornly reinforce failure, maybe they waste their time trying to patch things over with obstinate opposition.
This is a great opportunity to pull in some potentially less used skills. Drive checks to get from A to B faster. Empathy to figure out if this guy can even be swayed. Performance modified by persuasion to get a local union / trade group / business associate to join them. Look at your player's character sheets, find the skills they don't get to trot out as often and let them rock for once!
Just don't forget their opposition is likely also being proactive. Maybe the difficulty for various actions is set by the opponent's active opposition -- or maybe the opponent already met the politician in question, and has already 'set' a roll the players have to beat. Don't be afraid to roll that as if it were current, active opposition, then let the players spend FP to overcome it. Just let the players see you actively note their rolls, and let them realize that the enemy might come along and steal their precious votes away.
Instead of the meeting being the meat of the story, maybe it's the climax where they spend all the aspects they've been generating -- "Support of the local PD", "Councilman Bourroughs Disapproves", etc etc.
I might consider a multi-stage vote. Try to get it yanked from the schedule for 'further consideration' during the approve-the-agenda stage. Try to prevent anyone from seconding the motion to keep it from being discussed. And if all else fails... the emotional plea to the council not to pass it, as a final chance.
Don't be afraid to let the NPC cheat to keep up the pressure. That's what NPCs do. Maybe he's already laid a lot of groundwork (gained a lot of aspects) on the measure. Maybe he has a few stunts to make him a better talker. Gauge the players, gauge the story, and chose the right moment for the NPCs to hit hard -- and then let the players squirm and chase after a way to fix it.
And if a player spends FP, or is in his home territory... don't ever be afraid to let them be awesome.