Bruce. Thank you!
Gil. Thank you!
Regarding lubrication, for hot air engines....
I see a lot of differing ideas. And, I often try them out. Always returning to my old favourite. (Suggested by Reid, many years ago).
If it is metal on metal. Rotating, or sliding. I use 5w-30 full synthetic. As in my car.
The only variant is
this engine......
The displacer is moved, only, by the impulses from the power piston.
It will only run, with half a drop of
this oil on it's horizontal displacer shaft.

Most of my engines have graphite pistons. So, need no cylinder lubrication.
I never use graphite powder on power pistons, preferring an oil film to seal the running clearance.