You are in a similar position to where I was 2 years ago. Wife wanted the 3 rows of seats to replace our prior crew cab pickup, although far less capable than yours. An older Burb would have been a TV mainly, and newer would be a DD too.
In theory, the 6-spds are a huge game changer (and 8+ hitting the market now too). Despite having less engine torque, the gearing provides more wheel torque. The difference is having to downshift a gear or two to get it vs. the 8.1L and your diesel.
The reality is most people towing with the current gassers and 6-spds are cruising in 5th gear at 2000-2400rpm. Most hills can let the speed drop a little or downshift to 4th gear @ 3000rpm. That's better than prior times when they cruised at 2700rpm and dropped to 2nd gear at 4000rpm.
The 6.0L looks relatively weak compared to competitors gassers, even the half ton tops engines. But it is still strong. It pulls pretty hard over 3500rpm when you want to. It is darn near buulletproof, in production for about 15 years. It's no Duramax, but it does it's job very well. And GM has the highest % of gas sales in its HD trucks compared to Ford/Ram.
We put 21k miles in the first year on our 2011. Over 2600 miles non towing to Disney. 1k miles weekend wedding in TN. FE started at about 12mpg in my mixed commute, 15mpg on the Disney trip, up to 14mpg mixed commuting for my wife and 18mpg highway. I saw 20mpg over 200 miles on a day trip. One month shy of 2 years now, and only 34k miles. It's temporarily turned into a TV and all family vehicle for a little while.
Ride is about the same as my 2003 F-150. The kids and wife all pass out on trips. They only way they know we are towing is turning around to see it. It's easy to drive at or above ST trailer tire speeds, in 100* temps outside and 72* inside. With only 3 kids and 2nd row captain chairs, there is a lot of space for each person to keep the things to occupy their time 3-20 hour trips. With all the time we spend in it, we wanted all the conveniences and luxuries. It's not perfect; It just works. There are competitors with advantages in some areas, but not with the capability for RVing this thing has. If the receiver had 1200-1300 pound rating, I'd be a lot happier recommending it more universally as well as upgrading my own TT. Performance is underrating IMHO. It can whip my 6500-ish pound loaded TT around if I want to. I don't think another 2000-3000 would be a problem at all.
Based on your TT dry published weights and scaled weights, I don't have any problem recommending the 2008+ 3/4 ton Burb/Yuk XL to you. You will be right at the receiver rating as well as GCWR of 16k. But it will perform far better at that limit than any half ton SUV and pickup except a HD payload F-150 Ecoboost with not quite as harsh a ride and as much luxury as you'd like to purchase.