1) Sell T2i with 18-135 kit lens ($600 after Amazon cut)
2) Sell 17-55 f 2.8 ($650 after Amazon cut)
3) Buy Used 5DIII (LensRentals sale) or 6D and 40mm pancake ($2300).
Total cost of coverting: $1050.
I wouldn't want to spend any more money for a zoom (24-70 or 24-105), so the pancake would have to do.
So, what you would you do?
I'd go for a second hand 5dmkII, the 50 1.8 and the Tamron 28-70 f2.8 (cheapest) or the Canon 28-135.
Why have I gone for cheap lenses? Because viewing on a monitor or TV isn't demanding of IQ, while even A3 prints will be fine. They won't allow for aggressive cropping, but you're not likely to do this.
Why do I include a zoom? Because the majority of the photographs will be spontaneous rather than planned out or staged. The flexibility of changing focal length also allows you to eliminate much of the distracting background by standing further back and using a longer focal length.
I suspect you will use the camera a lot, so better lenses would make a lot of sense. The 50 1.8 is awful for manual focus, the 28-70 lacks IS and the 28-135 is a little slower than the 24-105. If budget isn't limiting you then stretch for the better lenses.
Would I recommend the MkIII over the Mk II? Not if it meant you bought inferior lenses. If you're moving to FF to get shallow depth of field a more expensive body crowding out the fast glass is absolutely the wrong thing to do. Also remember that we're nearing the end of the current generation of Canon's sensor tech. Future stuff will be much better - better than the stop or two between the mk II and III.
disclaimer: I don't shoot as much as I like and find focal length more valuable over depth of field. My budget is also limited. It's crop for me.