Actually, you do lose payload, BUT not off what your payload says on your label. Fuel is calculated to figure what your payload rating is. Some people try to calculate for towing and add a full fuel load to their calculations. You do NOT need to worry about fuel anymore since the companies agreed to use the automotive engineers standard here a few years ago... AND that includes a full load of fuel for specific trucks in the calculation.
So, this website will give you your tow rating and payload by VIN number
https://www.ramtrucks.com/towing/towing-guide.html#/landing/bytowing
Apparently, at one point there was an issue with this site, but it was "fixed" as of last year. So, put the VIN of the one you are looking at and this is it.
Edit: Oh, My 2021 has the 26 gallon fuel tank. I WISH they had ordered the 33 gallon. I had the 32 gallon on my 2015 and it was great, no negatives except the hit on my wallet when I filled it.
OH and what is with dealers who order towing technology options, including mirrors and the full Monty around towing and then don't spend the 95 bucks for a 3.92 rear gear or put on the 33 gallon tank. I just don't get it. Load it up for towing and then don't spend a few bucks for the guts of towing. Sorry, personal problem and rant.