I see the order of importance as (descending):
1. Tire and rim ratings as measured against the axle weights. Most of the camper weight is on the rear tires.
2. Physical axle rating (the actual axle, not the RAWR). Most of the camper weight is on the rear bearings.
3. Rear axle weight rating (RAWR) - Most of the camper weight is here.
4. Front axle weight rating (FAWR)
5. Gross vehicle weight rating (GVWR)
The gross (GVWR) is primarily a function of the tires, axle and frame/suspension. A dually and SRW of the same series (3500/350) almost always have the same frame. The rating goes up because of the tires. For a long time manufacturers capped the GVWR of SRW trucks at 9900lbs for registration and tax reasons. Throw two more tires on there and you get 11k+ rating. This is not a comparison of 2500/250 to 3500/350. That is a totally different topic.
I'm not suggestion or endorsing exceeding any rating but this is the list I considered when sizing my camper to my truck.
As far as "stability" of a dually, I've never been sure what that means. Most lean is in the springs/suspension not the number of tire. No doubt more tires increase number 5 above (GVWR) but they have little effect on sway and leaning or "stability." At least I have never experienced a difference. A big sway bar, air bags and good springs go a long way to limit leaning. If you need an extra tire out there to keep the camper and truck from falling over, you have bigger issues.
Certainly, if you are pushing the truck hard in a turn, more tires tend to give more friction on a dry surface and help keep the truck back end from swinging out. Try an over-sized trailer on an SRW in tight downhills if you want a high pucker factor. You can feel the rear want to get pushed sideways.
The labels on the camper and truck are a rough starting point but you have to weight everything - period. Everything else is a guess. The campers always weight more than their labels and the truck weight can vary too. It's hard to decide if a camper is "too much" before you buy it, but you can with some research get close enough to move forward or rule out a rig. Ask on the forums if anyone has actually weight there rig together and separately. You can learn a lot.
Scales - they are your best friend.
Edit: Added rim ratings as being as important as tire ratings.
