My guess is the tuner is providing the extra fuel for the higher EGT's, and the camper is causing a lot of wind drag then.
IMO, the 4" exhaust isnt going to be a "miracle cure", but it should lower temps around 150*. Maybe someone here would have a better tested figure than mine since mine was changed in stages.
Yeah, you definitely should have a boost mod like an elbow or j-hook depending on your turbo type to allow for more boost. Also a boost module of some sort to fool the ecm (usually come with most timing boxes). Or does a hypertech tuner take care of that with progamming? Also, hopefully, it gives a mild timing increase to help with EGT's.
I'm sure you are keeping the RPMs up over 2000 for flow, if not, you might want to give that a try.
When I had a pop up camper on my truck with DD1's & a VA box, using the stock exhaust with just the muffler removed, I pushed into some pretty good headwinds, but dont think it really ever got past 1100 much so maybe the gauge is reading a little high, or there may be a boost leak.
Might need to go to a bigger turbo housing as mentioned above, but there is a point of diminished returns there as well, and you trade lower EGT's for more lag and smoke on takeoffs.