I have the 2012 version of your truck. Our trailer is similar in size to the one you are looking at. With this combination we have travelled about 5 months per year for the last 3 years for a total of over 30 000 miles. We been to the Atlantic provinces, Key West and San Diego twice. Our setup tows very well.We do not get pushed around by passing transports at all - in fact the majority of the time I can't feel any effect.
With the Rear Living room all storage is at the front so tongue weight does climb. We have an Equalizer 1400/14000.I had a 1200/1200 for the first 2 years but switched up since the tongue weight was up at the top for the 1200. Both were stable.
The engine does rev on long steep grades but the Chev small block is built to do that.
I always use tow/haul ( and turn truck headlights on).
I think based on my experience you should be fine.