Being that your a manual it will do fine. If you absolutely want to re-gear I'm sure you'll enjoy the mod. If you'd rather spend the money on other mods I think you would be just as happy.

Personally I'd re-invest money in other mods that would benefit my jeep. Just my two cents

 

I think your right it's Christmas time and I need a rear tire carrier and front bumper or EVO Rock sliders

 

See that's kinda back asswards. If you're moving to 37's, even with the manual, I'd opt heavily for a regear. With 4.10's and 37's, you're asking your drivetrain to withstand a lot of strain while wheeling. The constant clutching gets kind of old too.

I have 4.10's in my 2011 2 door Rubicon, which I may add is fairly light, and I desperately want to regear it. Driving on the highway kinda sucks... I always have to be driving it, if you know what I mean. It doesn't posses enough mechanical advantage in the drivetrain to not suck and keep itself at a constant speed. I usually am bouncing between 95 km's an hour and 110 km's an hour... mind you we have a lot of hills around here. That's even with 35's that are running kinda small. I wouldn't dare bump a tire size without regearing it.

 

The argument of straining the drivetrain is a balancing act. A numerically higher gear will be weaker. 5.38 is weaker than 5.13 is weaker than 4.88 is weaker than 4.10. As you go up numerically, the teeth become smaller.

 

Why are you comparing your 3.8 to the 3.6 engine? The engines are dramatically different, the 3.6 isn't nearly as gutless as the 3.8. Although a regear would be noticeable it's not something that truly needs to be done. I've been driving on my 37's with 4.10's for over a year and it's been very live able, my jeep is my DD with quite a few hills / mountains in the area.

 

2013 6spd Ruby here... I didn't like 4.10's with 35's, can't imagine wanting to do it 37's.
Sure anything is doable. I don't like slipping the clutch. If you don't mind reving the rig up and slipping the clutch more power to you.
I am running 5.13's and 35's, it's a little high strung on the freeway, but all crossed up in the rocks nearly tipping over and trying to avoid dragging the windshield I'd actually like to have 5.38's to slow it down even more.

 

I've been running 37s with 4.10 gears on my auto 2014 JKUR and it is definitely doable, as a matter of fact I hardly know the difference now. Stopping distance was more noticeable to me than acceleration, I will upgrade my brakes before I re-gear.