alright so i own both along with 300, 9mm, and regular 223/556 ar’s.
i have taken four deer with the 6.8 and in my opinion it’s a fantastic mid-sized deer hunting caliber that’ll fit through an ar-15 magwell. none of those deer ran more than 20 yards. i hunt over fields that range out to 300 yards but i’ve never shot at one more than 100. the 6.8 is purpose-built for an ar-15. components from a quality manufacturer will work reliably but a couple of downsides: ammo cost and speciality bolts/mags. if you are shooting 200 rounds per weekend, it will be costly at ~$0.60 per round for the cheapest range ammo that i’ve found. if you reload, 6.8 is an excellent option because it uses .277 bullets i think the same as 270 winchester (i don’t load my own so this needs to be fact checked). there are a number of great factory deer hunting ammo choice available.
i also own an x39 setup that i recently built. range ammo is much cheaper but some mags offer less than perfect reliability. i haven’t deer hunted with it yet but it should probably get the job done with well-placed shots and specialty ammo. i wouldn’t trust my life to this rifle because i’m not confident it will go bang every time i pull the trigger. like the 6.8, it uses unique mags and a special bolt. you won’t find the selection of deer hunting ammo like you would for the 6.8 but they do exist.
for suppression - neither are going to be all that great with factory ammo. for a worst case scenario, i would rather have my 6.8 rifle for reliability but it would be costly to acquire the ammo necessary to stack high and deep. also allegedly 6.8 bolts will not last as long as 5.56 bolts due to “less metal” to accommodate the larger case. ditto for x39 bolts.
i own two 300 setups as well. you have the benefit there of standard bolts, mags, and components save for the barrel and muzzle device. also you can get much better sound suppression with sub-sonic ammo although your zero will be off versus the ammo you’ll want to use hunting. i don’t believe that the 300 even in supersonic loads will equal the ballistics you’ll get from 6.8 past 200-300 yards.
in a worst case scenario, i would want my 5.56 ar-15 or my x39 ak. i’m not sure either of the options you’re asking about would be my first choice.
these are some of my thoughts and i’m by no means an expert.