6mm BR which shoots the same bullets at the same speeds
Umm... NO, it doesn't. Where in the world did you come up with that? You're right that the BR is a much more efficient overall cartridge than the .243 Win, and the .243 Win is not a particularly efficient cartridge; that much is true. But your blatently false statement cannot stand. The .243 Win gets 300-350 fps more than the 6mm BR at any given bullet weight.
To answer the original question, I'd say, with a vitals shot, up to around 300 lbs safely, without any performance / penetration concerns,
provided you use the correct bullet for the job. With a neck shot or brain shot, probably around twice that, or 600+ lb ungulates, again, provided correct bullet choice (and as always, correct shot placement).
Oh, and I like the .243 Win, because it's the perfect LIGHT-recoiling neck-shot whitetail rifle round, which also serves very well as a target rifle round out to 300 yards. If I wasn't sure that I was limiting myself to neck shots on deer (but I do
when I carry the .243), then I'd choose a .260 rem or bigger - that much is true. But the .243 with 80, 85, or 90 gr hunting bullets (not varmint bullets) is the perfect laser-like, hammer-of-thor-DRT for neck shots on whitetails, and it's fun to practice with to boot.