The thing I don't care for with the interlocking type is they usually result in a thick tambour which isn't an issue on a roll top desk but is on a breadbox that may only be a couple feet wide at most. Since you can't use cloth how about cable? There were a lot of tambours made with fine cable stringing the slats together. The kind of cable used for fishing leaders would work.
Amazon.com : 49-Strand Cable Vinyl Coated 7x7 Stainless Steel Kit 30ft 175lb 1.0mm w/10 1.3mm crimps : Sports & Outdoors You might want to use canoe bits to make a cove and bead for the edges. If they become a going concern you could set up a gang drill to make the holes rather than individually. Monofilament fishing line itself might work but you'll have to test it to see if it has the longevity to do the job.
For the couple cloth backed tambours I made I had several boards going at a time to make the slats. After surfacing and thicknessing the boards to the width I wanted, 3/4" if I recall, I would joint both edges of the boards and then rip them on the table saw. Then back to the jointer for another pair of edges and then saw them off. I did it until the boards were too narrow to safely handle. The jointer and table saw were both kept running at the same time. You end up with slats with 3 sides surfaced and the sawn side was glued to the cloth. You could do the same and clean up the last surface in the thicknesser if capable of handling thin work or a drum sander. For larger production shapers or molders make more sense than routers unless you are using the interlocking bulb joint.
Pete
