This is probably of no use but my 2012 Jeep Grand Cherokee Overland with full air suspension came with Michelin 265/50 R20's. When they got worn down they wanted to hydroplane when it rained, so at about 30,000 I replaced them with the identical tire. I noted no change at all in towing (I do a lot), still very stable and solid, with absolutely no indications of any sway. The trailer is either a '74 Argosy 20' or a '14 FC 20. The hitch is an Andersen.
I have noted no towing difference if the water or holding tanks are full or empty, they are all behind the axle in the FC 20. In the Argosy the 35 gal water tank is up front. In other words, I have noted no change due to weight shift and distribution.
I have read of tires making a lot of tow stability difference, and you seem to have proved that possibility out. However I have no ideas or opinions as to why it should have happened to you with your combination. Is there any chance your tire dealer will exchange them for some other brand, given your experiences? I doubt that you will find a good solution by either tire pressure changes or moving things around. I would not want to drive a combination which was that sensitive at any rate. It would seem to be on the edge all the time.
I will be interested in how this plays out for you.