You should be more concerned about your payload than GCWR. Load up the truck and trailer like you're getting ready to start a camping trip with the family. Head to a CAT scale and verify that you're under all of your limits.
How to use the scales.If you're over any of your limits then begrudgingly use the more capable truck. Since you're only a weekend warrior making fairly local trips, if you're barely under or right at your limits then I would use the RAM. When you think about it not very many of your engine hours will be spent towing. It won't be easy street and you'll never forget that the trailer is back there, but it'll be manageable.
I know you said the tires were changed, but I didn't know if you meant just replaced or upgraded. If you haven't already then you can upgrade them to a P-Metric XL of LT tire. This will give you a little more stability and will help with handling. You can also replace your existing bump stops with Timbrens or Firestone Work Rites. They're basically bigger bump stops that engage sooner which helps take some of the strain off of the suspension. Sounds like you're truck's fairly new, but if you feel that your shocks are not up the task then you can always upgrade to Bilsteins. Stay on top of your maintenance and inspection schedules since you'll be towing at the upper end of your capacity.
None of this will increase how much weight you can carry or pull, but it will definitely help with the handling and comfort level. Before you go tossing money at the truck first verify that all of your weights are in acceptable ranges because if you're over then it's not worth it. No point of sinking money into a sinking situation lol.
