To the original poster:
First, a bit of background. This is what I shoot:
This is a custom 45-120 Sharps, built by C. Sharps, from Montana.
Specs on the rifle: 30" heavy octagon barrel; long range rear sight, front spirit level, front sight with interchangeable apertures; double-set trigger, and (important part here) pistol gripped shotgun style
stock. Done in a French Grey finish.
And, for the ammunition:
These were light bullet loads: 490 grain Lyman bullet. Now, I shoot three other bullets: 550 grain bore-rider with lube grooves, 550 grain spire point with lube grooves, and a 500 grain paper patched bullet. All of these are loaded with 120 grains of Goex 1F; compressed 0.60 inch, under a grease cookie (patched loads) and a sealer wad (.030 King's fiber wads, used for both sealer and overpowder wads.)
First thing--that stock you have will punch you. Consider getting a WIDER stock.
Next thing--it sounds like you are almost attempting to benchrest shoot the rifle. Don't do it! You want to pull that rifle in TIGHT. REALLY tight. Obviously, you so NOT want it to get a run at you.
I would also stop using the rear support bag. Put your support hand under the fore end, and use it to pull the rifle in tighter.
When I head to the range with my rifle, I usually shoot 40 to 60 rounds--because that's all the ammo I have with me at the time. Also, consider switching to 1F powder, or even 1 1/2F (Swiss) powder. It works very well indeed.
The idea of getting a PAST recoil pad is a good one. Best of luck to you--shooting BPCR is a ton of fun!