If it's only you and one other person and you don't go overboard on gear, you may be ok. But if you max out the weight, you'll notice it when traveling in mountains. If you start out with a trailer that's only 5,000 lbs. unloaded, you would end up towing around 6,000-6,500 lbs. and towing performance would be much better.
If upgrading your truck is an option, you could always try it and take the trailer to the scales and get an actual weight when loaded. Then you could see how the numbers crunch and either keep your truck or look for another. If you look at my rig details, you'll see that my Tundra is technically rated to tow 10K, but I really wouldn't want to do it. But with 6,500 lbs. in tow, it handles very well, even in the mountains.