Low profile, compact work-holding solutions. Advice sought.

[EddyCurrent]

I like Celtic. The old fodies way to stick something down for machining was shellac. You melt it using heat for the fixing, either melt or dissolve with methylated spirit for the removing.

In that case animal glue should work too, but these days maybe hot melt glue would work.
With double sided tape, I've discovered the strength seems to be a function of how hard you press it all down, in fact 3M I think call it 'pressure sensitive' tape.
With a job this small you would think the tape might have some sideways movememt that could be significant in proportion to the job size so a physical restraint such as a pocket would be best.
Maybe you could machine a shallow pocket in wood or perspex then fit the workpiece inside it, held down with double sided tape in the bottom.
Nice job by the way.