Short pieces of wood, with a long piece required for the top edge lip 
seems to be the problem.....

You COULD scarf the short long grain pieces into a long piece.....  DO 
this by using segments, each glued in place one at a time... the last 
piece is the tough one.... or as you suggested several short pieces of 
thin material, pinned to the top like a Shaker box.....  The only 
disadvantage is that you have expansion /contraction of the top ( which 
IS thick) being constrained by this lip..... which means something will 
explode.

or, if you are feeling ambitious.... Take all your short pieces and 
cross cut them to lip height.  Glue them up together into a long short 
grained piece....

This is then applied around the table top, in 1 piece with a small 
section left out...

You then make up the small missing piece to fit, and glue it into place.

This is not as hard as it sounds, and the short grain nature will allow 
it to be bent around the outside with minimal ( not zero) effort.

The biggest problem is the clamp around the outside... I have used strap 
clamps with great success..... My top was 20 inches in diameter.....

The THICKNESS of the piece is really important.... if too thick, it 
swill not bend.  But on this design, I suspect 1/4 inch would work.

If the top moves due to expansion and contraction, the lip will break.  
But the break will be more of a split.... as the short grain nature 
allows for this, and the lip will not constrain the top.

Frank Filippone
BMWRed735i@g...