A few things you'll want to consider here.
Yes an ax15 should be able to handle 37's and a 4.0l even if it's a stroker with no issues. It's talked that the original factory AX15 was rated for 300 ft lbs of torque while the AISIN ones available now are rated for 385 ft lbs. It's a very popular transmission for your application. We do make a few components to upgrade a couple weak points on the factory AX15 which you can find here https://www.novak-adapt.com/catalog/transmission/transmission-parts/ax15-parts.html that will help beef up that AX15 a little.
Now the biggest downside to an AX15 is it only has a 3.83 first gear ratio, but with your doubler you should get a pretty decent crawl ratio. But you can also achieve this by going to a transmission with a lower first gear. *That's if you're not needing the overdrive*. If you wanted to look into a truck 4 speed like a T18/T19 or a SM420/SM465 they you can achive that lower ~6:1 granny first gear ratio without even having to use a doubler.
Now we are releasing our NovaDrive in the next few months that would be the best of both worlds in that sense. You'll be able to use a super strong truck transmission with that granny low first gear for crawling and the overdrive for highway driving.
Those are a few things to consider as you are selecting your transmission.
Let me know if I can be of anymore help.