You asked for opinions, here's mine.
I've had my OIL on the Missile Range. I used a 300Win Mag. It was a 285 yard shot, and a good shot and he went down in sight, so with my three allowed helpers it only took us one trip off the side of the mountain to get to the truck. My youngest son shot a bull with a .30-06, again one shot and easy to find. My oldest son shot his with a .270. His initial shot was longer, required about a mile of tracking, and another killing shot. I've also accompanied 3 other hunters with their OIL hunts. One broke a spine and required a killing shot with a .270. The other two were one shot kills, both using my .300 and handloaded 180 Nosler partitions.
As mentioned above, their vitals are in a different place than North American game. They are very low and very far forward. If you haven't been on the missile range you can't fathom how much ground there is. Think 50 miles wide by 150 miles long. They can run a long way when wounded, and ethically you have to follow them. My recommendation is you break their shoulder, both of them if possible, and if your bullet is strong enough maybe you'll hit a vital too. Study the material Game and Fish sends you to change your thinking about where to hold.
So for me I would never recommend anything less that a .30 caliber, but that's just me. I have friends who have killed them both on and off the missile range with a 6.5CM, so it certainly can be done. But why go into a OIL hunt handicapped from the start? My suggestion to you would be to pick a .30 caliber gun that you can feel comfortable with and learn to shoot it well from as many positions as possible. If you asked me, I would suggest a .300Win Short Mag. And use a gel pad on your shoulder for the range days so you can put enough shots through the gun to feel comfortable with it. Also, the mesquite and other desert plants out there will likely not allow you to get a prone shot, you're going to be shooting off a tripod, or your backpack, or some other item that will allow you to stand for the shot.
Congrats on drawing a hard to draw hunt, and enjoy every minute of it. Chances are you won't get there again unless a buddy invites you along to help.