I have to cast my lot with the 300WSM CROWD. I am a 300/338 RUM and 300PRC nut but without a break and at 20-24" you are asking for pain and bad form. I'd seriously consider a 300WSM and maybe a proof carbon barrel. Id thread the muzzle 3/4-24. Put a cap on it if you don't want a break. I'm on my second right shoulder rotator surgery and at some point testosterone gives way to practicality. Besides, a shortish barrel is fine for the 300WSM. We actually build 300PRCs on 22" barrels that are designed to be shot with a can. This is the TEAM set up. There are countless options for you. Personally, I like to keep things simple with known (non exotic) calibers. I build a lot of 308 Wins for exactly this purpose. 260 Rem, 6.5 SAUM, 300 SAUM in 24" are also great options.
I guess it really depends on what you intend to shoot at 650. If its deer or the like a 308 win is hard to beat & ammo/components will always be around.
Just finished the FDE 308 with 19" Blake 6-groove Palma chamber, M2 (ish) taper. MDT TAC21 chassis. Built to shoot 168 SMKs. Will shoot first time this week.
The AR308 pistol has a 13" Shillen 1:10 match barrel we chambered and contoured. So far it shoots Fed match ammo at 1/2 moa.
The odd looking black bolt pistol is a 12-1/2" 308win on a XLR chassis w/a Blake barrel blank. 1:10. Shoots 1/2 moa. It was built to have an OMEGA can screw on and just stick out the guard. The can on it in the picture is a TB I think.
We break all our guns in. Unless the customer wants custom load development, everything is built to shoot one or two factory products. (Fed/Hornady match most often)
Short barrel builds seem to be gaining popularity.
good luck and remember, guns are like beer, one is seldom enough.