Same can I'm running on my 20" 30 nosler. This will be long, I load for quite a few shorter barrel long range guns.
As far as cartridge choice for a short barrel, decide the performance you want first. At a max of 700 on elk, and mostly deer, a .277 cal will work, though is on the low end, in my opinion. I built my rifle to be a 20", 1000+ capable elk rifle. For me, that meant propelling a 200-215 class high bc bullet 2900-3000 fps. So, I chose a 30 nosler. 208's at an easy 2940 with h1k
My cousin's wife's rifle I load for is a 19" 270 Sherman, essentially a highly improved .270 winchester. It shoots 140 badlands solids at 3060, and 170 Bergers at 2760. For comparison, their 25.5" rifle sends the 140's at 3240, and 170's at 2940.
I would, however, without a doubt go with a properly throated 300 wsm, on a mid length action (huge difference from SAAMI spec). You should be able to get in the 2750+ with a 200ish bullet without pushing things, there is great (adg) and good (norma) brass available, tons of bullet options, 300 wsm's are easy to get shooting well and stay in tune, there is tons of data available, and recoil won't be bad at all. My 9.5 lb 30 nosler is comfortable to shoot supressed, and a wsm will be better.
If 30 cal isn't your thing, then my only other consideration would be a 7 SAUM or 7 MAX (SAUM improved, can still shoot factory SAUM ammo). We have two 7 max's, both 20" tubes. My wife's 7 max is sending a 170 PVA Cayuga mono at 2835, with a conservative load of H1000. With N565, 180 Bergers will run about the same velocity. My father in laws 7 max is sending 180 hybrids at 2775 with n565, on a lower node, with a proof barrel that only has 25 rounds down it. I expect it to pick up 30-50 fps after fully broke in.
On a side note, if you go 30 cal, I would strongly suggest going with an 8 twist 4 or 5r groove.....mono bullets are improving quickly, and there will be some 180ish grain options from a certain manufacturer coming out that will have the bc around 210+ grain c&c bullets, and I have high hopes for terminal performance. I'm sure others will start to follow in suit as well. Badlands also currently has good options ranging from 175 up to 205. Jus sayin'.