All the rifles you currently own will work just fine. That being said, I understand that you may NEED a new rifle for this hunt. I think just about everyone on this site has that addiction and I'm no different. I purchased a Christensen Arms Classic in 338 Lapua for my mountain goat hunt this past October. Love the rifle and it performed flawlessly. You may want to look at their line of rifles. Mine only weighs 7.1 lbs bare. If you want a 6.5 cal, they have the Creed, 6.5-284 Norma and the PRC. Jeff Brozovich (who is very well respected in the shooting world) has been giving high praise to the 6.5 PRC. He has one in their Ridgeline models. I personally asked him how he likes the rifle/cartridge, and he stated it's been the easiest rifle to dial in that he's ever had. He hasn't even gone to hand loading for it yet, because it shoots the Hornady ammo (ELDX & ELDM)so well. I read a recent post from someone that had a friend that worked for Christensen, and he said that the Mesa LR models were consistently the most accurate of all the models they produce. So, food for thought.
I personally own a 6.5-284 Norma. It's a tack driver and recoil is like my .243. Gives your more velocity, energy, and distance than the Creed. The PRC is it's ballistic twin.
Now, I have a friend that just booked a sheep hunt in Canada with Steve from Steve's Outdoor Adventures TV show. He's hunted with him in the past, and this was a cancellation hunt. Steve has sent my buddy his personal Begara rifle chambered in 6.5 Creed to practice with and use on this hunt. Again, food for thought.
Choices, like yours, are first world problems. Figure out if you want a new rifle or not. Look at the ballistics of the different cartridges, and make the best choice for you. If you go the 6.5-284 Norma route, look at Cooper and Christensen. The load data for the 6.5-284 are severely low since it has been a wildcat cartridge for some time before Norma standardized it. With a 140/143 gr bullet, you should be in the 2950-3000 fps with a 24-26" barrel with a near max load.
If you're going to buy factory ammo only, then the Creed or the PRC are your best choices.
One last bit of advice. Train your *** off for this hunt. I thought I was ready for my mountain goat hunt, only to learn I was very wrong. Go to the local high school football stadium and do the stairs. Once you feel good with that, add a backpack with weight. Then do interval training where you go as fast as you can for 30 seconds, then normalpace for 2 minutes, the fast, etc. It's the only way to really get prepared. You can't prepare for altitude, so you just have to be in the best shape you can be. I had a brutal 7 hour hike in and 6 hour hike out. It's a grind!!
Good luck.