Towing with a motorcycle in the bed?

I'm likely doing an out-of-state move soon, 1250 miles.

Plan to put a BMW R1200GS (~550lb) in the bed along with a few mountain bikes and miscellany (~200lb). And probably 400lb in cab, including weight of myself.

Truck is theoretically 5700lb wet, with a 7050 GVWR. So 1350lb of cargo. I'm at 1150lb summing the above.

Add to this a 4000lb (est. loaded weight) tandem-axle Uhaul trailer, 10% tongue weight (400lb).

I assume my payload capacity decreases 1:1 with tongue weight? Or is it more than that?

If it's 1:1 I'd be at 1550lb on the truck itself, which is 200lb over GVWR.

I can probably try and keep as much weight out of bed as possible to lighten rear-axle weight, by putting stuff only in cab and trailer other than the motorcycle.

Obviously I won't be hauling ass, my experience is that Uhaul trailers tend to go unstable a bit north of 65mph, even when correctly balanced.

This isn't going to be a regular thing, but don't want to ruin the easily-flattened springs.

My gen1 worked fine hauling two Yamaha FZ1s (1000lb) in bed, two people and a lot of gear (but no trailer)...was about the same amount over GVWR and had no issues. Have yet to haul anything heavy in gen2.

Truck is a 2019 crewcab, stock.

What do you all think? Will it be riding on the bumpstops?

You'll be fine. This country was built on people overloading stuff and getting the job done. Seriously though, if it's just one trip and you're not in a particular hurry, I wouldn't worry about it.
Now if it's laying on the bumpstops in the driveway, you may want to move some stuff around.