Another couple things that might hang up your synthetic stock swap is the bolt handle and the rear sight.
The old 721/722/725 has a straight bolt handle while the 700 is swept back. A short-action 700 stock will fit but you'll have to do some work to the bolt handle relief groove to fit the stock to the bolt.
I believe all of the 721/722s had the band-style rear sight. You'll need to inlet the stock for this too.
I don't know of any aftermarket stocks for the 721/722 quirks so at least these two mods will be on the list. I'm not sure about the bottom metal...I've never tried swapping 700 and 721/722 floorplates so I don't know for sure if there are any differences.
As for bobbing the barrel, I'll have to agree with Diamondback. Unless you're drastically shortening the thing up you aren't gaining much weight savings. and if you bob too short then you'll start losing muzzle velocity. I would shorten enough to recrown if it's damaged but otherwise I'd probably leave the tube alone. If you do shorten the barrel and muck up the crown it is possible that you'll take an accurate barrel and turn it into an inaccurate one.
Is your 722 a 24" or 26"?
Around my area, there really isn't a big collector following for the 721/722/725. Prices are right on par with a similar condition 700. If it's a really mint condition piece then it would be worth a bit of a premium, but if it's any lesser condition I don't think modifying would be considered "ruining a classic". If it is any higher grade (premier, peerless, special, etc) than the standard, I would hold off modifying it also. The collectors do prefer the higher grades and that will up the value too.
I don't want to steer you away from modding your rifle into something that suits your needs but it might be more economic to start with another rifle that is more setup for your needs. A Model 7 in .223 perhaps sounds pretty much identical to what you're proposing to build with your 722.