It’s not ideal. I’ve killed 23 and been witness to 50 or so being taken. A big ram can take a beating to put down. Also the vitals are more forward than typical American game. You’ll have to favor the shoulder more than normal.
Not at all saying it you can’t do it, I just wouldn’t venture below a 6.5/264 myself.
If you’re very confident with your set up to deliver a precise shot at distance you can always break them down with the neck/shoulder shot.
Wouldn’t recommend. Three of us went in January and killed three mature rams. First ram took a 199hammer hunter at 3128fps from my 300 win mag through the shoulders at 225 yards. and hit the ground. Ram 2 took a 160 accubond out of a 7mm mag at 445 yards. Bullet went in around the last rib and went out the opposite shoulder. Took two more hits to the boiler room and neck at 100 yards to finish him. Ram 3 took two 200 lrx out of a 300 win mag at 330 yards. We have all killed bull elk, mule deer, antelope, etc and we decided Aoudad are a special kind of tough.
It will work with a good placed shot. I have been present on about 100 harvest. I watched a very big ram roll off a side of a hill with a well placed 95 grain berger out of a 243 WSSM at 285 yards.. I have also watched a few 25-06 and 257 WBY hammer the crap out of rams to 500 yards..at 600 the combo above will be on the light side of steam. If I were you and wanted to use a 243 I'd try to stay in the 300 yard range. I like 6.5 and 7mms to 600...I really like 30 cals and 338s...but hey it will kill them..just place your shots carefully...if you want to shoot Aoudad past 400--and want to stick them--try a 338 Norma with 250 berger EH ....does wonders...
So what was the point of asking us our opinion? Also yup a varmint bullet to the neck would do the job. I was only going to get one shot at this so I stand by my personal opinion of .243 not being enough gun for Aoudad. unless perhaps you have lots of opportunities such as living where they live but for a guy like me who may never get to hunt them again I’ll stick with a magnum and put it in the shoulders.