Mike you asked about the veneer earlier on. It came from Rutlands - about £20 for 3 ft2. It's probably less than 1mm to start with and I then took off a bit more when sanding out the adhesive that had seeped through.
Thanks Terry. Apart from proper stabilising, I think veneer is the answer. After spending a few wasted hours on two sets in amboyna and red burl I now feel confident that the veneer liners add the necessary stability.