If I ever come back to the 'Land of Enchantment' it will be the Ghila. Having said that, I'll probably come up to Albuquerque and pay you a visit....
Lock time on an in-line ML with 209 primer ignition and in my case Blackhorn 209 is so close to a centerfire you won't even notice. You'll like the recoil. Even with a huge pill, the recoil isn't nearly as abrupt as your 338, in fact, it's a very mild but pronounced 'push' with really no muzzle climb even though you can get the high pressure/velocity ML's with muzzle brakes now.........
.............You can actually purchase ML's that will chrono at 3000fps with a 50 cal saboted or unsaboted pill. That makes those very capable rifles to 500 yards and even better depending on conditions.
You might also want to step away from the saboted pills and look at non saboted pills.
The reason why some States outlaw saboted rounds is the plastic sabot never degrades. It's there, on the ground forever.
The latest 'Hunt'in Fool' has a big article on ML's and the western states various requirements.
I think NM is unrestricted as far as sights (scope or iron) goes and unrestricted as to ignition (some states don't allow in-line 209). In some states, the primer must show partially when the action is fire ready. Some states don't allow optics at all.
Here where I live, ML's are standard fare. We can't hunt with a centerfire unless it's a pistol. Shotguns, bows, pistols and ML's.
I have a Bergarra barreled CVA Accura V2 thats clearance shimmed with a Nikon 3-9 and a BDC reticle and because there is no restrictions on saboted pills here, I launch a Barnes TMZ ballistic tipped pill with 100 grains of Blackhorn and a CCI ML 209 primer.
I prefer Blackhorn because you can load repeatedly without the sabot/pill getting tight from fouling and cleaning is a snap with a patch and bore blaster.
Opening day, I'll be on private land a couple miles from here reducing the overpopulation of whitetails and stocking the freezer.
I've shot a couple different flavors of ML's and the in my humble opinion, the CVA Bergarra barreled out shoots everything I've shot. Matter of fact, CVA offers a money back guarantee on it's accuracy.
Out of the box they have a couple easily fixed issues. One, the breech plug to breech face space is too large on some rifles (due to production tolerances I assume). That causes the primer to lift slightly at ignition and allow hot gasses to leak around the primer cup and erode the breech face, easily fixed by shimming the firing pin retainer out to become a tight fit to the primer cup and the trigger needs to be adjusted slightly. No big deal in an inexpensive (compared to a centerfire rifle) hunting rifle.
ML's tend to be nose heavy compared to a similar centerfire but I don't notice that as a rule.