Here is where things get a bit dicey.
The max payload is a function of axle rating, suspension rating and tire rating. The axle is a non-issue as it is the same in both the SRW and DRW. That leaves suspension and tires. IMO, suspension is likely not an issue, but if it is that can be corrected with suspension enhancement of some sort, so not a real big or expensive issue. Tire.....you may be able to find a tire that has a higher load rating to carry those few extra lbs, but in all honesty, I don't think that would really be needed. I'm sure your tires are already load range E, and rated to handle 80psi cold; check the side walls to verify.
The big question - Can you upgrade the suspension and tires to carry the load safely? IMO - YES. HOWEVER.....right from the factory your truck has a rating from the manufacturer, and it doesn't matter what you do, that manufacturers rating will NOT change, so if you end up exceeding that rating (safe or otherwise) the law will only see those numbers compared to what they measured.
In the real world there is little chance that any inspector is going to even give you a second look if everything 'looks' proper. That includes the '3500' sticker on the side, the 5th wheel setup AND tires that 'look' round even on the bottom. LOL so chances are that you will be able to enjoy your travels without enforcement issues.
BUT......If the unforeseen happens (ie. Accident) and they starting checking in-depth and discover that you were running in excess of the manufactures rating, then you could be in a world of hurt with regard to law suits, insurance issues, etc.
BTW - Do you actually have the trailer already? I ask this because I believe the listed pin weight may be listed as the maximum if the trailer was fully loaded. I just seems a bit high from my experience, but then again my experience with newer travel trailers is quite limited.
You are looking to reduce the pin weight by about 350lbs to be within specs, so getting the actual pin weight may prove to be rewarding. Also there is the possibility of adding extra weight behind the trailer axle(s) to reduce the running pin weight. As that is likely where your water tank(s) are located, adding 1000lbs of water may be the ticket to get all the specs in order.