Mark,

Places like Rocklers sell materials to fill this gap. The string idea
would work, too. Your idea of putting a strip of wood will work. Except
if you plan on using a machine planer be very careful. The opposite
grain you will be presenting the blades on a surface that may or may
not be glued very well under your glue job may just send splintered
pieces everywhere and ruin your panel. I suggest adjusting the
thickness of the attachment wood before you apply it to the panel.

Roy