I totally agree with you that the lab question is the most important question about the validity of the liver flush.
If you are interested I can email you a reference of a very old method for analyzing cholesterol. This method was developped in the fifties or sixties of the last century. Thus, it canot be an very sophisticated method.
One point to mention:
If your lab will oly put your stones under a microscope and look at it your stones might lack the typical crystal structure of older gall stones. Therefore, I would really look out for a lab that performs cholesterol analyses.
Since you live in Italy maybe I can help you through some connections to the Milan university. However, I am not sure and do not want to promise anything.