It seems to me like your focus is having something fairly light weight, in a handy size, for doing short range work on a relatively tough critter. Were it my decision to make, I'd consider acquisition of another rifle in a caliber that won't be handicapped by a short barrel, while also being well suited to suppression...and choose an appropriate projectile for the task. While not a fan, and don't own one, the 300BK seems like just the ticket in this case...or maybe 7.62x39.
The 243 is a speedy round, using a healthy dose of powder to push light weight bullets at a fast clip. While it may be possible to tune the round for a short barrel using faster powders with a heavy bullet, it's still going to be slinging lead at high velocity with a large volume of gas. You may or may not get the expected performance from your suppressor. All in all, you're trying to hobble a race horse for plow work. I'd say, if you have room in the barn, buy yourself a sturdy mule and save the thoroughbred to do what it was meant for.