If you are worried about recoil I would look at a semi-auto action. From my limited experience I would say moving from bolt to semi-auto reduces recoil about 15% and if there is extra weight, there usually is, another 10% is not abnormal. The action not only absorbs some of the recoil, but also spreads it out over a slightly longer period.
You can get the rifle ported or purchase a ported model. Remember, the more aggressive the porting the louder the rifle will be to those around you.
If you hold the rifle correctly a 308 out of a normal rifle should not give you a flinch. If it is hard to handle I would look into how you are holding it. A super light sporting rifle may kick a bit though.
I would try to shoot before 6 seconds b/c at that point my balance and sight begin to be effected. Not drastically until 10 seconds or so, but there is an earlier effect.
I wouldn't have anything lumpy moving around under my shirt shot to shot. There are sew on recoil pads for shirts. Many good shooters use them.
Here is an example although I am unfamiliar with this design/brand.