If you are going to try and shoot 1000 yd br, that means 30 cal in all probability when you add hunting with it. With the 30, it seems that the 300 Weatherby in tight neck .334-.336 or improved Weatherby of some type would be best. In competition, brass quality is key, so Norma brass is available. Get 1-10 twist barrel 28-30" if it will make weight. If $ is no object consider fluting to give you that extra inch or two.
Recommend you have built to PA 1000 standards which is 16.5 lbs straight taper to .950 or .940 with muzzle brake
added, that way it is legal for IBS or NBRSA matches also. They have 17 lb limit and no taper requirements. Any gun that meets PA standards will meet the others, but not necessarily other way around.
Large capacity case such as 300Ack Imp or 300 Weatherby will shoot 240s very well if you want. Gives you great round to hunt with also.
Stocks McMillian Tooley MBR or Bruce Baer laminate. Cheaper and straighter than Bill Shehanes. Both 3" front end.
Barrells- I am partial to Clay Spencers in VA, or Hart. Lilja is good also. Spencer builds some of the tightest tolerance and best barrels I have seen. Seen too many non-shooting PacNors, Kriegers and K&P to take a chance on them. Personal opinion and bias, take it for what it is worth.
Recommend you sleeve the action if using a Remington. Try speedlock kit from Holland Gunsmithing with lightened and oversize firing pin and stronger springs that will not kink.
Finally, find a good 1000 yard BR gunsmith. Someone who can really and build and get a competitive 1K BR gun. Not just a "parts assembler" who has not built competitive guns or shoots competitive. John Myer of Custom Gunsmithing and Clay Spencer in VA, Bruce Baer in PA, Leonard Baity and Dave Tooley in NC.