As noted, the 165gr. bullet weight is a good one, but for heavier game such as elk you
will be better served by going to a 180gr. Don't worry about the slight (very slight) difference in trajectory with the heavier bullets - you won't lose enough to really matter.
And hey, like you said, you're not exactly an expert at 300yd. shooting. As far as that goes, not many shooters are any good out to that range, even assuming their equipment is up to the task. Under field conditions 300yds. is a long shot.
So, check the ballistics tables if you like, but you'll find not much difference out to that distance. I'd recommend the better or premium bullets for elk, such as Nosler Partitions, Trophy Bondeds, or Remington's Core-Lokt Ultra. All can be had in the 180gr., as component bullets for handloading or in factory loaded ammo.

From the Remington ballistics tables:
LONG-RANGE² TRAJECTORY*
Cartridge Type Bullet 100 150 200 250 300 400 500
Remington Express 165 PSP CL 2.0 1.7 zero -3.4 -8.7 -25.9 -53.2
Remington Express 180 PSP CL 2.1 1.8 zero -3.5 -9.0 -26.3 -54.0