I ran 37's on my 08 Jeep Unlimited. I did have 4.88 gears but it had the 5 speed with a crappy 1st gear. The 3.6 has more power and the 8 spd has a really low 1st gear. Plenty of newer JKU are running 37's on stock 3.73 gears with the 6 speed and say it's bearable. Curb weight of a fully outfitted JKU (bumpers, full size spare, winch, skid plates, etc) gets up around the 5200 - 5500 lb mark so they are comparable to the Ram. Expect it to be a sluggish of the line, but it should be bearable.
I think you'd be happier with a gear change, but not happy with the price. Probably looking at $2500 or more and you can only go up to 3.92. A year ago, the front axles I found for sale were about $800 - $1000. The rear gear change with install ran about $800.00. If you can do the front axle swap, then your looking at about $1800 - $2000.00. If not, then you get closer to or above the $2500.00 mark.
good luck and post pics of your build when you decide.