assembled pic in gray for metal effect
While looking at the Stirling engine design, I wondered why you needed a cooling piston. Room temperature air should be cool enough. That and if you have a pressure release valve you can release the pressure in the cylinder. So this is my test subject. When heat is applied to the base, the air in the cylinders expands. As one cylinder reaches the top of it's cycle, the hot air and pressure is released. causing the other cylinder to have greater pressure and for it to rise. One concern with this is if the air only has one out, eventually you'll run out of air in the cylinders to cause a difference in pressure. So I added 2 small holes on the inverse side of the release holes, so hopefully the cycle will raise the piston enough to reach here, and as the air is escaping the larger hole hopefully it will be drawn into the smaller.
4.762 w x
2.54 d x
1.875 w x
1 d x