too many options right now. What desirable characteristics are you looking for in your rifle besides accuracy?
Weight limits, type stock (wood, glass, thumbhole), how do you hunt, reload or do you want factory capability etc, do you like cut or button barrels, looking for max velocity in a barrel (canted rifling of Broughton, Rock etc).
BATs and Stolles are BR actions and not for working hunting rifles to drag thru the woods IMO.
I would not make them my first choice as they are just a little too tight for a reliable hunting gun. BATs will and have locked the bolt up with one grain of sand on the lugs. I know, mine did and had to take it to a smith and repolish the lug after and set it back. BATs are great but softer metal than other actions too.
I would look at Stillers Predator or TAC 300/338 as my choice. Stiller top end price is $400 under cheapest BAT and $300 under Stolle, and readily takes all Rem parts, mags and triggers. These are proven quality rigs, no questions and proven delivery times are very reasonable.
Pick your barrel, Hart, Shilen, Broughton, Lilja, Krieger, Brux, Bartlien etc. Choose first cut or button, regular or canted rifling, that will tell you what barrel mftr to use. Holland QD brake or his new one would be my choice. They are proven, quality, designed for hunting rifles, and not super expensive. Max 28-30", cost no object order it from the mftr fluted.
Stiller action, Wyatts box and I would use standard tuned Rem trigger to 1.5 to 2 lbs. Lower pull is not needed or recommended for true hunting gun. Save $140 cost over safetied Jewell. Now if you just want to say I have a jewell, go for it but not needed.
Caliber, 30 Cal is a given as minimum for distance, KE and bullets, maybe 338. 300 RUM, 30-338 Lapua and 338 Lapua. you can use the TAC 338 for the Lapua case which is 10x better than Rem so no need to go lesser quality Rem based wildcats which are driven by need to stay in a standard Rem action. How important is availability of factory ammo? if important, that makes 300 RUM and 338 Lapua top candidates.
I would order my own no neck turn reamer and go guages from Dave Kiff at Pacific Tool and Guage or Dave Manson at Manson Reamers.
240s in the 30 and 250-300s in the 338 will go in excess of one mile accurately with enough KE for what you want.
Stocks- too much variety. What do you like? Wood, glass, thumbhole, adj cheek etc. Joel Russo, McMillan, Manners, Holland, HS etc are all good and you can find one that fit your desires and needs once you define them a little more.
Stiller 20 MOA base ordered with action, pick your rings. TSR rings are 1/3 cost of NF or Badger and just as good. No need to spend more.
Scope- Leupold, NF NXS or Schmidt and Bender, all three will work well, some better than others but at an increased cost.
Weight depends on hunting on foot, horse etc. I would try to limit it to 12-13 lbs with scope.
Smiths- lots of very good ones and certainly not limited to this board. Find a local one if possible. Nothing like being able to drop it off, and talk to them in person eye ball to eyeball. Look here ( we now have 4-6 good smiths here) and at "top recommended smiths" on www.6mmbr.com
in particular, has highly recommended smiths all over th US that have 30 yrs building top quality LR hunting rigs like Bruce Baer, Mark King, John Myer, etc.