I have the same rifle as you are asking about. Rem 700 Tactical .223 with 20" bull barrel. Mine sets in a HS Precision, full aluminum block, stock. I use a Caldwell 6
-9" pivoting bi-pod and a Badger Bone rear bag rest, or shoting sticks I made from dowel rod and surgical tubing, depending on the setup.
I can send you pics of groups with 75gr AMAX that are .63" @ 100 that I hand loaded on Dillon 550, so in a quick fashion, and still deliver.
The only problem is that you can only load 1 in the mag and 1 in the tube because they do not seat far enough.
69 BTHP will be just as good and you can put 5 in the mag and one in the chamber.
62 Gr American Tactical (made in Turkey ironically) have delivered consistent .48 or less groups and can be purchased off the shelf or on line. I have not fired groups over 200 with them yet, but at 200 the worst 5 shot group so far has been 1.16"
Good luck and good hunting!!