I am thinking of a Norman sack of Rome as a POD, Avignon becomes the heart of the Western Church on this scenario causing slight split between the Roman pope and Avignon.
Avignon wasn't a Papal fief until 1274, more than two centuries after the Schism. And if you did that you wouldn't really have a United Church, would you? More likely the Western Bishops break off when Rome condemns the Filioque as heresy (if the Byzantines keep a firm hold on the popes this will probably happen, plus I know how these reunion TLs usually go) and elect their own patriarchs/anti-popes to give their barbarian overlords some legitimacy. This communion would probably be far more volatile theologically and in my opinion would experience further Schisms down the line.
True. But does my word choice make sense in the context of our discussion?
It does, but I think that the difference between how the Church is run and matters of Faith would be seen as a false dichotomy at the time (and still is by hardcore Catholics and Orthodox). It would all be seen as part of the Deposit of Faith, and the only reason I'm being so nit-picky is any realistic way to avoid the Schism wouldn't have either side treating matters that way.
I like the idea. It probably has its own problems, but they seem less likely to drive "East" and "West" apart and more just individual patriarchs (counting the pope as a patriarch for my convenience), which can be dealt with as such.
So just like it was before the Schism?

Well, the pope still was a patriarch until Benedict XVI dropped the title back in 2006, and even then he still functions as one for the Latin Church. In my opinion, we'd still see regional heresies and Schisms, but not ones that couldn't be dealt with. ATL theologians would probably fill reams griping about "Toletism", or "Gallicanism", or "Hibernianism" and how they were all out to corrupt the faithful

.
Although, if the West can hold together, we may see antagonism develop along a North/South line, since most of the Christains in the Middle East, Egypt, and Ethiopia were Miaphysite. If Islam is butterflied away, then we could even see the Nestorians taking over the far East and all the alt-Albigenses running off to the New World once it's discovered to found a "Western" church.
Just think, a sect for every Cardinal direction!