Can you post your whole circuit or planned circuit? There are several members on ETO who are very good at doing simulations with LTSpice. Even I can hobble along a little with it. That may save you a lot of time and effort on the breadboard.
As for the 555, information about triggering is in most of the datasheets, but there is an application note from Philips that I usually go to, because it is easy reading. It also has a few little tidbits that are not in many datasheets. The note is AN170, but last time I checked, Philips had taken it down from its site. I uploaded it to ETO here:
https://www.electro-tech-online.com/attachments/555an-pdf.52952/
If you are still concerned about continuous triggering, then you can AC couple your trigger as shown in Figure 3 of that note:
View attachment 83069
How big is your Stirling engine going to be? I used to belong to a Yahoo group, but it seemed mostly talk and demonstration. Very few people made engines that could do work. That was a few years ago. Has anyone gotten to 1/2 HP or more with a reasonable size?
Sorry to go a little OT, but I do find heat engines interesting.
John