Don't be afraid to look at online suppliers for tires, just make sure that you've got someplace that will install tires purchased elsewhere.
On the subject of going to a higher load rating...
Keep in mind that if you get carried away on increasing your load rating you may exceed the pressure rating of the rims, for example: If you currently have Load Range E tires installed (80psi max pressure) you probably will not be able to use Load Range G (110-110psi max pressure) without replacing the rims with ones rated for the higher pressure. The same could also be possible if your factory tires a Load Range C and you attempt to mount Load Range D or E.
I would suggest that you weigh your trailer axles "before" replacing the tires. Make sure that you do this with the camper "fully provisioned" if possible, including a full fresh water tank even if you don't travel that way. The GVWR won't tell you how much weight is going to actually be on the tires. Most large truck stops will have a scale, also keep an eye out in your area for a DOT scale. If you find a DOT scale and it's not open a lot of times they will have the scale readout pointed towards the window so you can weigh it without anybody there, besides DOT scales won't cost you a dime. While you're weighing the camper do your truck, both steer and drive axle as well. You might be surprised to find how close you are to maxing out your truck drive tires with the camper attached if you don't have a dually.
I think that if you weigh this camper you will find that with 5k axles on it, Load Range D tires are going to be a pretty good choice.