There’s no such thing as “ELR Benchrest” in any common or national format. Lots of ranges do their own “no format” matches, and what’s best for those would depend upon whatever match design and rules they choose.
Sanctioned Long Range Benchrest will typically shoot 600 and 1000yards, such as what
@South Prairie Jim shoots. For this, there tends to be a couple of different camps - the 6BR guys, the 300wsm guys, and the 284 Shehane guys. But they are ALL shooting big bellied single shot actions in stocks with big flat forends and flat tails to track true on the front rest and rear bags.
National ELR matches are belly matches fired with super-magnum rifles, again, typically single shots, but they’re fired from tripods (Phoenix preferred for everything but Ko2M, Accu-Tac for Ko2M by a landslide). I provided that equipment list in your last thread - only super-magnums play here, even the 338 Lapua really gets left in the dust. BUT… these aren’t shooting just to 1 mile. Typically these matches are reaching out to 3000-4000+ yards. A lot longer than you’ll be able to shoot if your range only expands to 1 mile. These are typically dominated by 375 Cheytacs, 416 Stroker, and 416 Hellfire or Barrett.
For your 1 mile max distance benchrest game at a local range, without knowing their specific rules and their specific range limitations, a 338 Lapua would probably play very competitively. A 300 Norma might also play pretty well. But you’ll need to know the caliber restrictions of the range, caliber restrictions of the competition format, weight and/or length restrictions for the competition, and any other rules they might invent, because there is no standard for 1 mile benchrest competition by any existing sanctioning body.