I have killed elk with 3 auto-loaders in my life.
3 with an M1 Garand, (30-06, loaded with 220 grain RN bullets)
1 with an FN-FAL (loaded with 180 grain Nosler Partitions)
and 1 with a Benelli R1 in 270 Short Mag (loaded with 160 grain Nosler partitions)

The elk killed with the M1 and the FAL were all shot at between 20 yards and 60 yards in heavy timber.
The one I killed with the R1 was killed at just over 400 yards on the open prairie.

No problem with any of them.

A Browning BAR in 300 would do all you'd ever ask of it. Use a good bullet that will hold together and I'd bet you'll never have a problem.