I'd leave the 3.73s and try it out first. I had a 1999 F250 7.3 with a 6-speed and it had 3.73s and 35's. I never hauled a camper with it, but I towed with it more than a few times and I never felt like rehearing it would help anything.
It did great at 70 mph too. For reference, I also have had a 1996 f350 cc/lb and it had 4.10s and a 5-speed. It was okay, but if I remember correctly it was tacking over 2500 at 70.
These diesel motors have so much grunt that it doesn't affect things all that much if you throw bigger tires at it.
These things been said, 35" is the max is probably go on the stock gearing. After this, and maybe even with this tire change, you'll be affecting transmission reliability.
I currently have a 1997 cc/lb Dodge Cummins with a 5-speed and 3.55 gears. I have a lot of plans for it. But my camper probably weighs 3500lbs (I think would be a close guess), and it has 285s on it now. It does fine. But 35" tire height is the eventual plan, and the way it handles things now, I don't foresee myself wanting to re gear.
There are a lot of reasons to change gears, but your truck would be just fine IMO of you left it stock. Especially if your camper is going to be as light as a Four Wheel brand one is.
Edit: I just carefully read the post and want to ad this...I also have a 2006 Dodge Cummins that has 35" tires and a built automatic transmission in it. The truck has 3.73 gears and would like to lock/unlock quite a bit when towing over very hilly terrain. This is one of the reasons I looked so hard for a manual in my 1997. I'm a truck driver and didn't particularly like the lack of transmission control when towing with the other truck. But it never shifted out of OD or unlocked the torque converter on the highway.
Another thing you can look at doing for more control is installing a torque converter lockup switch.