A third axle will do what you need it to and I think that when towing on the highways, with such a competent tow vehicle you will not notice any issues in turning. You will drag tyres around corners but for the amount you use the trailer and tyre wear, that should be an acceptable price to pay. So long as you don't leave tyre marks on the driveway at home.....
Every triple axle trailer I've ever been associated with (3 of them now) was a tire shreader... Each had a number of cases where the tires would have ply separations and/or blow-outs. Each was running trailer specific tires and well within the load rating. One trailer was a Tag and the others were Goose.
I think for your application Ron, The best bet would be the relocation of the rear axle to a location forward of the front axle. It might require the relocation of both axles.
Also - Keep a close eye on your hitch. I've seen two Class V Hitches now (Reese) start to fail and one on a friend's 3/4 ton Suburban failed altogether with my trailer on the back.
Poor welds. But not all of them are so - so you need to inspect carefully and replace or beef up the hitch..
The hitch that failed could have killed my pal and his two friends in the Sub. As it was it totalled out the truck, trailer and Corvette inside.