A Camp Bed, 1803

This camp bed is found in Sheraton's 1803 The Cabinet Dictionary, which describes the construction under Camp Bed, starting on page 123. This is an elegant design that requires no technology unknown in the 14th century.

When trying to figure out how medieval people solved a particular problem and you don't have enough information, going back to the earliest era when you do have enough information is often a productive strategy.

Update: here's a picture of one of Washington's beds folded. This may be a reproduction rather than the original. You can see that that instead of offsetting the side rails horizontally when folded, this design has them hinge slightly upwards or downwards so that they clear each other. And here's good picture of Anthony Wayne's camp bed, showing how the sacking bottom is secured by rope passing through eyelets and over pegs.