It will be the 338 Lapua. Just plain proven and criteria calls for two large ammo mftrs. USMC sniper instructors had the Brit para AI 338 Lapua guns for quite a while and loved them.
That leaves out the RUM, Edge etc
They are already shooting the 300 Win mag and goes to 1200 or so, that means the 300 WSM, 7-08, 7 short mag etc are not even in the running.
As far as cost, it is only 4 smart bombs or one fighter engine for the entire USMC and Army.
Forget a 700 action. They are wanting robust action with long term reliability with multiple barrel lifes. That leaves out an opened 700 that many consider marginal.
It will be a bolt action, simply due to the accuracy over a semi auto at 1500 plus.
My bets are on the AI guns. A super 338 master LR BR smith (who will go unnamed at this time and some here have even used) has been testing different reamer and barrel configurations for them. He will be hand checking the test guns. I am sure by the time they get the test guns for the services, the rifles will be absolutely tack drivers and butter smooth.