Does the 722 have a swelling in the barrel for the mounting of a rear sight in the dovetail?
The BDL magazine has a little tab for a single screw that attaches the magazine to the bottom of the receiver, this is so the box magazine doesn't fall out when you open the hinged floor plate. The 721/722 is missing the small threaded hole in the receiver for this screw. The 722 bottom metal won't fit the BDL stock without some work, I would recommend using 700 bottom metal.
You may have to remove a small amount of wood for the bolt handle, on a 721/722 the bolt is straight down, a 700 bolt handle is slightly angled back toward the shooter. Also, the safety is a little shorter, you'll have to remove a small amount of wood for it.
I considered exactly this on a 721 (long action) but I wound up putting it in a nice ADL stock (no hinged floor plate). I had to inlet the stock for the above mentioned swelling in the barrel, remove a little wood for the safety, machine the metal box magazine shorter to fit in the ADL stock. I used a 700 ADL trigger guard and no shimming was required. Aside from minor stock in-letting the 721 fit like a glove in the 700 stock. and I know the 722 will also, I just haven't gotten around to that one yet.
I wound up putting a 700 bolt in my 721 so i didn't remove any wood for the bolt handle. Overall it was a nice upgrade for the old rifle.