How to map the polysurface so that the joints do not appear in the texture?

This may help, I’d use Unwrap with a symmetry tip and then CageEdit in the UVeditor to adjust. It looks like you were using a custom object but that will cause some overlap in this case due to the small folds in your target polysrf. That approach could work too with some editing to the single custom surface but I prefer this way…