Very good points made, and I found myself in an almost identical situation with our Sierra T22. A little smaller than yours (I like your floorplan a lot BTW) but pretty comparable.
I started with a 2001 F-150 4x4 Crew Cab...it was my hunting truck, but not a great towing truck. 4x4, lift kit, oversized tires all decreased the payload and towing capacity. Anyway, my F-150 struggled anytime I climbed and had similar gas mileage to what you've reported. I also had some issues with sway that a WD hitch helped, but could still tell the trailer was pushing the truck around a lot. I was honestly never really comfortable towing.
Had some mechanical issues with the F-150 (maybe towing caused some of the issues...unclear since it had pretty high mileage for a gasser anyway) and decided to trade up. We ended up with a 2011 Ram 2500 CTD. Could not be more pleased with this improvement. I know it's too much truck for our trailer, but we went the route of buying for our next trailer since the family is growing.
Night and day difference between the two, especially climbing. I've towed a couple of the steeper routes here in AZ with no difficulty and even ample acceleration uphill. No significant increase in MPG...maybe a couple MPG on average, but the towing experience is much better for me. I only use the Ram for towing and hunting trips with some very short in-town driving, so daily MPG and fuel cost not an issue for me. Only on the road.
I have a friend with almost identical trailer and a gas F-250 and he's very pleased too. I never feel like the trailer is pushing the truck anymore, and towed in some pretty serious wind and dust storms this weekend with no problems.
Honestly my favorite feature on the diesel is the exhaust brake. I rarely even use the brakes going down some really steep grades. Exhaust brake combined with the proper gear make for nice towing.
Just my two cents in what appears to be a similar situation.