Ok, let's take a step back here. The transmission is not shifting properly. Do we know why? It could be the larger tire size. Can you put the stock tires back on to rule that out? If not, get an AEV ProCal (or something else) to recalibrate the transmission to the correct tire size and gear ratio. Depending on the tools your mechanic has, perhaps he (she) can do that with a scan tool - don't know.
The transmission not shifting at the correct shift points can definitely cause your engine temps to elevate. But so can the warmer weather and driving up Floyd Hill, Eisenhower tunnel, etc.
The 2013's run stupid hot. Mine saw 240 degrees on certain grades at altitude. Are you trying to hold 70 MPH on the climbs?? And if you are solely relying on the temp gauge in the dash, you are wasting your time. Its not analog at all. It has certain presets that it goes to and stays there until a threshold is crossed - up or down. If you really want to know what your coolant temps are, you need something like an AeroForce gauge or Traildash. ( I think the traildash can reset your transmission for different tires and gears as well.)
Did your mechanic put the proper coolant back in the system? (It should be purple.) If not, that could be the cause of your issues, easily! This is a big flag - make absolutely certain he used the genuine Jeep coolant for 2013, not for 2012.
The 2013s run hotter than any of us would like to see. But after having mine gone through by the dealer under warranty (flush system to remove casting sand, then replace water pump a few months later) I am convinced it is just the way FCA programmed the damn things to run. And I believe Motech has pretty much confirmed this (but going off of memory of a post by Robbie).
If your dash gauge is running between half and 3/4 or even over at 3/4 sometimes, welcome to the club. There is nothing really to be done about it. You can't (reasonably) install an extra fan. You can't override the fan controller to turn it on sooner. (Although Robbie has hinted that he has cracked the code for doing just that. I just don't know if it is something he would provide on a non-LS converted JK.)
I don't think your winch has anything to do with it. The grill, who knows, maybe. But let's look at the obvious things first. The wrong coolant could be the culprit.