I don't believe this issue is repairable... I've had the release button 'stick' in its depressed position and I was able to use a pin to carefully release it again but I can't see how it could be reinstalled if it falls out completely.
The best fix is probably just going with the thick panerai spring bars like the version Panatime sells:
Spring Screws for Panerai - Set of 2
panatime.com