You don't need to increase rim size to get what you need. A 185/80/13 LR D tire is rated for 1710 lbs each, that's 3400 lbs per axle. Your camper maxes out at 5K?, it's a tandem axle.... So you can keep your 13" rims, install 185/13 tires and have 1800 lbs of capacity in reserve, not counting tongue weight which further increases this reserve. I think you'll be just fine, most trailers max out their tires / axles have little or no reserve. You could even keep the 175/13 tires and still have a comfortable reserve. Don't over think this.
Adam