We do something similar by towing a 6,400 pound F-150 behind a 2015 Newmar King Aire with 600 HP. Here are some points to keep in mind.
If you are towing 6 bikes, you probably are also camping with children. I'm guessing at last three kids. That means you will want the super crew version of the F-150 with a 5-foot bed rather than a 6-foot bed you'd get in a standard cab. You will have a hard time finding a super crew F-150 under 5,000 pounds and it probably won't be possible if you add any weight to it such as bikes.
Towing 6 bikes in the 5-foot bed is difficult, but not impossible. We currently carry 4 bikes in the back of ours and could easily carry one more. If we crammed the bikes together 6 would be possible. But here's the most important part, you will need to remove the front wheels off of the adult bikes in order to get them to fit in the 5-foot bed. Alternatively, you could build a frame on top of the bed and carry them above the side rails. This would give you room underneath the bikes. You could also carry 4 in the bed of the truck and two on a hitch behind the truck.
Loading and unloading your pickup truck will take about an hour.
As for performance, forget about it. Motorhomes are very slow to begin with. Adding that much weight will slow you down that much more. It's not a big deal climbing mountains as you can always take your time, but the problem is accelerating onto the highway in the first place. Believe it or not, there are old highways where the on ramp is of limited length and there are no emergency lanes at the merge point. I've had some close calls where the two tractor trailers on the highway in both lanes are physically not able to make way for me and I have no choice but to stop short of the highway on the ramp. Thankfully, that happens very, very rarely, but be prepared for it.
