I would start off by replacing the scope base with a quality one piece base.
I doubt that is the problem, but you could not give me a 2 piece scope base
after I saw what they did on a Browning 243 BLR. Darn thing shot all over the place.
After I put a dednutz mount on it, it now shoots tight.

1" to 2" sounds like it could be the bipod. Are you preloading it ??
I would take the bipod off for now and shoot off a front and rear rest.

If it still shoots open, I would try some black hills match ammo next
and maybe verify the scope.

How about paralax ??
Sure yuou got that set correctly.
Seems like I remember My ss scope was off in parallax

If it still shoots open I would at least bed the recoil lug to make sure the recoil is not being transferred to the action screws.
You may want to do that first or just bed the entire stock.

Last ditch effort, I would shoot a box of 50 tubbs FF bullets through it.

After all that it should shoot 1/4" 3 shot groups if everything else is good.
I think 10 rapid shots will probably open up some due to barrel heat.

Somewhere in there you might want to try some Berger or lapua Scenecar known accuracy loads.