"I have a question. You are in pursuit of an elusive pneumatic rifle, all stocks of which SEEM to have been concentrated within the borders of the Tyrol and all stocks of which were destroyed in 1809? "
No apparent question here.
"What arms manufacturer - of any age - is going to sell his wares ONLY within a small, mountainous province, with a relatively small customer base? "
Circa 1750-1800 pre-industrial revolution in these areas (like Tyrol) most production was for local consumption. These "manufacturers" were all small local gunmakers with few employees. It is interesting to note that the one pre-1800 Cortina airgun found so far is located in a Serbian museum, so, these airguns did travel.
"He / they must have hawked them around other places, in particular the Austrian military, certainly the biggest domestic market. "
No. Girandoni, once he was in Vienna, was bound for life employment to the Emperor and he was under a strict order not to sell his airguns to anyone other than the government. The Girandoni was considered a state secret. For this he was supplied with a guaranteed income.
"Troop trials would surely have been carried out and records of these trials would exist. "
There is a fair amount of info on the activities taken by Vienna government, trials, etc. See Haller 1891 "Die österreichische Militär-Repetier-Windbüchse" but that is no the issue at hand. What I have been trying to do is find information on the historical use and mfg of airguns in the Tyrol.
"I imagine these guns would have been relatively expensive, due to the high machining standards required, further limiting the market."
A standard Austrian musket cost between 4-5 gulden. A Girandoni complete with tanks, pump, and other kit cost well over 50 gulden, so, well over 10 times the cost. The airguns also required specially trained mechanics to work on them. So, yes, much more expensive than a musket.
Girandoni`s guns fell out of favour with the Austrian army because of the high cost of maintenance and repair. How could a Tyrolean peasant-hunter afford this extra burden?
The airgun never fully fell out of favor. The Austrian Genie Corp kept airguns for fortress defense purposes as late as 1870. Airguns have special qualities that cannot be replaced by any firearm. The precise reason why they were popular with the Tyrol hunters is not known and can only be speculated. Some of the possible advantages are: quiet shooting (valuable for a poacher), works without any difficulty in the rain, fast second shot, no gunpowder needed (the Tyrol is a collection of very remote areas, it is possible that a Tyrol hunter who had little access to population centers to obtain powder would prefer the powderless airgun) The Tyrol are famous for being intelligent and mechanically inclined and keeping up maintenance on an airgun is not that difficult for a sharp individual willing to learn and get their hands dirty.