Once again, I must say I really like Tim's ideas. I fully endorse all of them. 
Thanks. It's as though I've done this professionally. 
How about the triumvirate as it were consists of the two consuls (from Harco and Magvillus), and the Praetor, where Verdi has been campaigning for a third consul position for their house as the third domus magus- in fact some probably believe that the move to change the Domus Magus to Thebes is a political ploy to bring pressure to that effect. Other covenants are divided between hoping getting a consul position for Verdi will help get the proverbial foot in the door for everyone else, and worrying that it will just create a larger and more stable block of 'elite' covenants. Verdi meanwhile has been kept out primarily because they are unwilling to surrender their popular vote for the Praetor position. Harco tends to be in favor of giving Verdi a third consul position, since the Verditius are dependent on trade and this would strengthen their positions, while Magvillus is against it as it would dilute their power without providing any significant alliances.
Alternately, the above was the case, Verdi just got the consul position then started to move their Domus Magus from Verdi to Thebes and now things are really heating up since there is now no 'special' reason for Verdi to have a consul position while denying other covenant's their own.
It may make sense that what was an effort to share power with Harco and Magvillus has recently been stretched to Verdi, and is creeping toward a general consulate. (To which conservative magi say "See? See? A slippery slope!')
I am strongly of the opinion that Praeconis should not be historically elected in Rome. There may be a movement to shift to an elected Praeco, but there are two other nearby Tribunals with that habit. Rome is one of the central tribunals, and should have the customary custom.
The advantage of a mechanically selected chair is that it does not rely on popular opinion, a fickle and arbitrary, favors traditional magi who respect custom (a virtue, medievally), and should change fairly often as magi lose their grip on life or reality.* I imagine magi point to the election of bishops and popes as an example of manipulation and politicking.
That said, there is an election traditon in the Church and Italian communes and republics.
*Although, in reality, it favors those with a significant advantage in Arts or blood or making deals with powerful spirits or faeries.
I seem to recall (can't think of where) that the Roman Tribunal was described as "violent." So I'm wondering if the paucity if vis has something to do with that.
I dislike the idea that Rome has a "proper" system for distributing vis. If it's going to be a strongly central and Traditionalist Tribunal, then vis belongs to those who register it, or to those who claim it through certamen or Wizard's War. Which means that outside of the vis claimed by the domii magnae and the Redcaps (Redcaps being effectively immune to having their assets seized by force of arms), vis will often be fought over, leading to a tribunal where Wizard's War is more common than elsewhere and a more violent culture is embraced than, say, Thebes.
Even if the praecox isn't elected there is still voting for hearings at tribunals and changes to the peripheral code, which by their nature Harco and Magivillus would not vote on, so losing their right to vote for a fractional veto (which may be granted in case of either 2/3 or unanimous vote of consuls and praeco) would be a small price to pay. if Verdi has acquired the same rights for being a Domus Magus without surrendering their vote that would be a big deal, and if they then decided to relocate their domus magus...
I was amused by this variation, so I decided to educate myself: 
- praeco = herald, auctioneer
- praecox = premature, precocious, pre-cooked
Violent and somewhat squabbling makes sense to me, given the cultural background. I doubt Wizard's War is as common as Hibernia, but it could be more common than average. Certamen might be fairly common - I don't know when duelling culture developed, but it was once a widespread duelling culture all across southern Europe, and this could be similar in Rome. Furthermore, I recall a Tremere covenant in Rome, and they would certainly encourage certamen. Add to that an undercurrent of proxy fighting, cloak & dagger back alley struggles, and you've got quite a brew of intrigue.
A proper system of distributing vis seems unlikely. Harco has a stream of vis passing through in trade, and they would be against a system that undermined the market. Verdi is also production and trade oriented, so they would be against it, too. Magvillus' income is probably somewhat based on fees and fines, so they're perhaps neutral. I can see patronage, but you make a good point about a Theban distribution system.