That's a pretty light rifle for target shooting. I have 5 guns in .308 and the recoil varies a lot. Two are M1As which are heavy guns and don't seem to bother me, possibly because the action absorbs some of the felt recoil. One is a heavy barreled GAP Crusader (.920 all the way to the muzzle) that gets my attention but not overly so. I can shoot it all day but I'll know it when the day ends. One is a bull-barreled Savage 10-FP that gets more of my attention because it isn't as heavy as the GAP (but it's markedly heavier than yours). The last one is a black gun from LaRue Tactical (7.62 OBR) and that thing kicks the bejesus out of me, or at least it did before I put a big muzzle brake on it. It's just physics - the heavier the gun, the less the perceived recoil.
I also have 5 guns in .223 and none of them kicks at all. Older guns with slower rates of twist in the barrel won't give you the range you talked about shooting because the slower twist won't stabilize the heavier bullets needed to reach out there. But the fast twist barrels in newer guns can handle the heavy bullets - 69, 75, even 80gr. In competition, these guns do very well even at the 600yd targets provided they can use the heavy bullets. A twist of 1:8 or faster is needed for these, compared to the older guns with 1:12.
A popular trend in target shooting today is to go with guns between 6mm and 7mm. Some of these are pretty much wildcats (6.5 x 284) while others use older cartridges like the 280 Rem or 7mm-08, etc. These guns shoot lighter bullets than the .308 that do well downrange because they have inherently better ballistics than the 30 caliber bullets, yet fight the wind much better than the .223. And they kick less because they are throwing lighter bullets. The problem is that guns in these calibers are more rare (often custom guns) and the brass is also rare (often formed from other, more popular brass).