That displacer piston is the opposite, the copper should be on the bottom for max heat transfer, and stainless on the side to minimize heat transfer. Stainless is a poor conductor of heat. If you are daring. I would replace the bottom with copper, about .032. I used type M copper pipe, cut about 3" long, annealed several times and rolled flat with a large metal tube. How thick are the walls of the copper displace tube? If thick enough you could machine a heat dam near the top. A heat dam is a groove in the diameter. My wall thickness was .065, and I took it down to .030 for about .100 length. What this does is slow down the heat transfer. The thinner the material the slower the heat transfer. There are pictures of all this in the pdf, page 51.