I have the same curved table. While we almost never use it as a bed, and its almost always up, I too had the same issue when "trying it out". What I did was cut small "boards" like the ones attached to the dinette seats that hold the table top in the down position, to attach on top of them, making them "bigger" so that the table top could not slide out.
I suppose the proper way to do it would have been to remove the existing wood and replace the entire strip that was there, but I decided to do it this way. I stained them to match, counter sunk the screw holes slightly and ran wood screws through them and the existing pieces every couple of inches. It has worked fine since.
I have however seen a much better way (I think) since this time. Assuming yours is like mine with the large metal supports in the middle that can be removed, take one of them to the hardware store and buy some PVC the same size as the ends. measure from inside the "hole" on top to the inside of the one on the bottom and cut the PVC to fit.
This way, when your table is in its "down" position, it will also be supported by those legs, and not just the strips of wood around the dinette. You can store either legs under one of the dinette seats.
2020 Ford F150 5.0 V8 SCrew 4x4