You’ll want to confirm, but I believe 2013 had a front axle center axle disconnect (CAD). You can’t maintain the function of a CAD and have a LSD in the front insofar as I know. When the CAD is open, the front carrier doesn’t rotate, and the spider gears spin instead, with the left and right spiders going in opposing directions. No LSD I know of will tolerate that.
Power Wagons use a selective locker in the front, you could do that. Likely ARB or others make lockers too, but I’ve had no reason to look into aftermarket.
Another option I believe is deleting the CAD, then you can use LSD’s such as Eaton TrueTrac, assuming they are made for your axle. The downside on that is the front diff and driveshaft will spin all the time, so there’s a small mpg penalty and some increase in vibration (more so in lifted vehicles with altered driveline angles).