I doubt it's an air leak. It'll either take a lot of cranking to start or it will start up like normal and have a hard stall shortly after. There may be more things that could happen if there was an air leak, but that's been my experience with two different deuces.
You could simply crack the bleeder valve open above the secondary filters and let some air in or drain some fuel from the primary and secondary filters, not much but 5 seconds worth of fuel while that bleeder valve is open. This will let air in. Try to start it, it should start but it should stall once that air pocket hits the IP.
Honestly that doesn't sound like the issue, if there's an air leak, it will happen every time, cold or hot. I think the plunger may be suspect, but it's inevitable, parts in the IP will wear over time.
If you can't get the parts to rebuild the HH, from my understanding, the Chinese made HH which was talked about here on other threads seem to work pretty well and have a better system in which that button doesn't fall off like it does in the OEM HH if I understand that correctly. Last I saw those were around the $500 range.