It sounds like the Scimitar Oryx in Texas usually top out at around 400 lbs, only stretching to 450+ lbs on occasion.
And it sounds like shot distance depends upon the particular ranch that's being hunted. Some places, the average shot seems to be 100-200 yards. Other places, that jumps to 350-450 yards.
I wouldn't have any reservations about taking my .270 Win with 140 gr Partitions. It's accurate enough. The bullet is tough enough. I know the trajectory. And, most importantly, I have more experience with it in the wind than any other load (or any other rifle, for that matter).
From my point of view, the target is a very large mule deer, or a small elk. It shouldn't require artillery to take one down.
.30-06 would be fine.
.338 Win Mag would be more than enough.
If I was feeling froggy, I might even drag out the .35 Whelen. I'd just have to be mindful of longer shots and do some work beforehand to get more familiar with the trajectory.
I mean no offense with this comment, but can't come up with a nicer way to say it that still carries the same tone:
If you don't think .270 Win is good enough for the job at 400 yards, it would be idiotic to take a step
backward to 7mm-08.
The difference between .270 Win and 7mm-08 can vary, depending upon the particular loads chosen (and especially if handloaded); but you're generally looking at a 200 fps to 300 fps velocity difference for same-weight, same-style bullets. You can boil it down to the 7mm-08 essentially being 7% to 9% less powerful than .270 Win, with that being obvious in muzzle velocities.
If you like the cartridge and want one, that's great - go for it. But this particular argument just doesn't work for me.
7mm-08 is to .270 Win/.280 Rem what .308 Win is to .30-06.