Tow capacity is a moving target. Is that number specific to your truck or did you get it from a sales brochure. The same truck can have different tow ratings depending on the axle ratio, cab configuration, bed length, etc.
There are lots of different types of trailers: boat, flatbed, horse, dump, utility, cargo, camping, etc. Of them all a hardsided RV is probably more difficult to tow than anything else. They have a huge frontal area, huge side "sail" area, a high percentage of tongue weight, and not much ability to move weight around. Your truck might have an easier time towing a 6,000 pound boat than a 5,000 pound RV.
Also, as you add weight to the truck the tow capacity drops (the way most vehicles are rated) pound for pound. So, if you add 500 pounds of people, pets, cargo, and truck accessories to your truck that weight comes off the tow capacity.
Your 4,400 pound dry weight will be more like 5,500 when you get all your stuff in the trailer. Dry weights normally don't even include propane, batteries, water, or even dealer installed options like an awning or AC unit.
Even if all the weight numbers work I suspect when going over mountains you will have the gas pedal flat on the floor, RPMs around 4,000 or more, and running in 2nd or maybe 3rd gear. Is this "struggling" some would say yes, others no. It won't hurt the truck but you may not enjoy the ride and the overall longevity of the vehicle won't be that great.