I've been shooting my 7mm-08 with 140gr ttsx for a bunch of years now. Killed whitetail, mule, blacktail deer, black bears and elk with it at ranges from 8 to just over 300 yards. I also shoot heart/lungs and get the same results. Full pass through,excellent wound channel. Usually a short death run and they drop. The only oddity is black bears. They all seem to drop on the spot (again heart/lung shots) "no complaints here".
My buddies daughter has done the same with the 7mm-08 low recoil loads with 120gr TTSX.
OP, a guy I work with shoots TTSX's from his 300 ( not sure what powder/load he went with). I know he's taken whitetail's a moose and a black bear with the same load (I never heard him say anything negative about any of them). He did recover a bullet from the moose from the offside shoulder and it was pretty impressive.
That said, IMO your 300WM with ttsx will work on pretty much any north american big game animal with allot more reach than I'm limited to with the 7mm-08