Try a dot of double sided tape. A strong one.
Another thing to try is to take a plastic tube like a pen and shape it to grip the outer edge of the part.
I assume that the part is the inner ridged button and that it has some lateral motion, most likely downwards? Have you poured some naphtha or maybe ammonia in the outer collar to get some liquid to the base of the part? I expect it to be like the lef-timer lever on later 'Flexes- a ridged button with a tube going down which is threaded on the end. The down tune is less than 2mm in diameter. The threadeds go into a flat lever. In this case, pulling it down resets the counter, yes?