7mm-08 is pretty ideal in this role. It retains more velocity than the 6.5s while pushing higher B.C. bullets at that lower weight than the .308. It's a bit of hair splitting, but it really is a pretty optimal suppressed short barrel hunting round. That's why it's my first pick for integrally suppressed builds, which have ported 18" barrels. The effective barrel length is ~15-16", the 9" core makes for an OAL like an unsuppressed 26" rifle, and they put down at ear levels in the neighborhood of a suppressed .22 Pistol while still managing 2,500-2,600 FPS with 140-150 gr bullets.
This is my demo Sussurrus, built on a Mossberg Patriot 7mm-08. It's 8.5 lbs with scope, just took a mulie with it a couple weeks ago
View attachment 1117626
A muzzle mounted can obviously won't be as quiet as this integral, but a good one will make the mild mannered 7mm-08 very pleasant in the field.
The .308 would be my recommendation for the non-handloader, though.