Mountain Towing

Put it in tow haul and don't look back. If the tranny starts hunting in the hills drop it in manual and select the gear level that does the lowest RPM with the least hunting. Your truck shd do just fine. I think you will be about 6000lbs loaded with a TW of about 750lbs.

On he up hill side, gear down and keep you RPM's up. Down hill I will drop my speed about 10 MPH at top of hill and drop down a gear or 2 as required to let engine help control descent. If speed gets to high, I apply brakes HARD to drop maybe 5 MPH below speed limit. This will let brakes cool if you need them again. NEVER ride you brake.
Have traveled 30K in last 7 years all over the West mountains,and even Virginia and West Virginia mountains and never had a problem.
I have a 5th wheel and use a 40+ year old Hydraulic brake controller. But with a bumper pull I would recommend a good sway control. You can probably find more information on sway control and brake controllers In the RV net.

Don't know where you live, exactly, but I live in "far" southwest VA and we pulled our 3500 lb 17' long hybrid with a 2001 Chev 1500 with the 5.3L for several years. It wasn't the fastest thing on the road, but it pulled it OK. We would NOT be doing 65 mph when we reached the top of the mountain, but we weren't going 40 mph, either. We would slow down to 50ish on the long, steep grades (like Christiansburg, mile 122).

I have no recommendation on the brake controller because I really don't remember what we had on the truck.

How did you find a "brand new" 1971?