I would try to use low viscosity glue. In Germany we have "Schweisskleber" which translates to welding glue. But it has nothing to do with welding. It's just a low viscosity glue. It comes in these cans
Deflate the mattress and use a brush to generously apply the glue along the seam. Let it dry and repeat. Do at least 2 layers, recommend 3.
I normally fasten a few of the buttons/poppers of the duvet cover before loading it, this usually stops anything from getting inside the cover itself.
As for a large item tending to gather smaller items into a ball, I think this is something I used to see with old-fashioned machines decades ago, but any machine designed in the last 20 years will normally ...
What you want is a clothes horse.
also known as a valet stand. You can find them in many different designs and materials. If you want to make it a permanent fixture of your bedroom or dressing room, you can surely find one which matches your decor.
In a top loading washing machine, put in two similar duvet covers on opposite sides of the drum, and nothing else. In a front loading machine you might be able to do just one duvet cover (and nothing else). Wash your sheets and pillow cases in a separate load.