Very interesting stuff you are playing with.
If you are connecting or not connecting the beam, is really a matter of how to extract the power, and should give some insight into what really affects timing. Timing is not used as much as I personally think it needs to eventually in Stirling engines. It looks like you are beginning to unravel that.
Also, the cold end ...
By pure theory, make the cold end colder should make more work. In your case you did not see that.
One thing is that your liquid nitrogen is not really cooling the gas inside the chamber. Here you have hit upon two issue.
a. you need to thermally seperate the hot and cold ends. any chance to buy a small piece of Space Shuttle tile on ebay? :)
if all you are doing is fighting your hot end with your liquid nitrogen, it may actually make things worse, and that is SO not right.
you may also need to find a way to get the extreme cold into the chamber center, besides the engine casing, which is already being heated at the other end.
a cooling coil thermally insulated at the entry and exit points?
b. the cold end is suppose to collapse the volume of gas, to prepare it for expansion on the hot end. well, at the very least, it needs to do this. even at this, the engine is unbalanced. a good Stirling design will not only push on the hot side, but pull on the cold side. many engines run on only the expansion on the hot side, probably most. this is a drag in many ways including requiring more flywheel to get past the low power spots in the curve. if you want speed, you will want to lighten the weight.
But, yeah, you need to focus on playing with the thermal cycle, before moving on to methods of extraction and engine RPM. You will only be able to remove the power level produced.
Please keep us informed as you go.