I don't know squat about suppressor's but if the thing recoils to much I'd say get a new rifle! Suppressors do two thing's for sure. First they have to change the balance of the rifle hanging the weight on the end of the barrel. Then, IMO, they take what might be a good looking rifle and turn it ugly looking! You said, I think, You only have the one rifle, get another rifle in a lessor cartridge firing a lighter bullet even if it's the same case. Or try a recoil pad! L just can't help ti think the extra wight on the front of the barrel is gonna change how the rifle shoots and handles. new rifles today are spendy but so is gunsmithing. But a rifle that discourages you I humbling at best. I had a 243 well two of them. Never hunted deer much with one even though I had good results with it. had my 700 barreled to a 260 Rem with a 22" Lilja barrel and it's handy as ever and to 260 hoots the pants off that 243! Down side is it cost my $800! Took an extra 25-06 Rem in a Rem 700 action years ago and had it barreled to a 6.5xx06 with a Shilen barrel. it also cost me $800 but since doing it I've never reached for my other 25-06 again except as a loaner. If your not a fussy shooter, I'd suggest you trade the 308 in on a 6.5 Creedmoore or a 7mm-08. Lighter bullet's is gonna cut down on recoil some and if you get a lighter rifle, also cut down some. Then again if it's not recoil but noise you want to control, ear plug's aren't that expensive and should work well. have to admit 308 is an all time favorite with me, but if I wasn't happy with it for some reason I'd disown it! My 260 is not only pleasant to shoot but it is also now one of my most accurate rifles using 140gr Speer HC! have never shot a 7mm-08 but shot a 7x57 a lot years ago and that 7x57 with 154gr Hornadys wasn't bad to shoot at all so I suspect the 7mm-08 isn't either.
Turning shooting into a chore doesn't help increase enthusiasm much. If its just the noise, most anything you get is gonna come with noise and you will probably end up wanting a suppressor so just get ear plugs and give your 308 another chance.