Noting wrong with a Bipod.........
I use one all the way out to 800 yards with stock 270 and 30-06 hunting rifle.
However the non-trigger hand is not on the hand guard and moved back to support the trigger hand and stock area to keep stock movement to a minimum off a bench table.
You don't need a Gold Dot "Best Choice" 168 gr.bullet out of a rifle to have a load with enough energy for a person for SD.
A 125 or 150 gr .308 bullet is plenty for any in home to 300 yard work, in your rifle with a proper bullet design but if that is all that you have right now, go for it.
However you might want to try some lighter weight bullets that have less recoil than that 168 gr bullet, if you can find any and maybe save a little money.
As for a custom trigger, yes they will help.......
but you should give the stock trigger a chance and maybe get it polished if you have to, before going the route of a new trigger.
Lots of shooting time will teach you where your trigger takes up the "Slop" and when it will "Break"...... use snap caps if needed, for dry fire practice to get those cross hairs "Fine tuned".
The difference of a normal shooter and a Great Marksman is.........
time with his rifle.