"...with 22 balls in the tube, one in the transfer port and one in the barrel, that adds up to ?"
Note: a ball in the transfer port is, by the very nature of the design, in the barrel, so, there is only one possible addition to the magazine count not two as your post suggests.
Thinking about it, magazine count should only be the number of balls that the magazine (and the magazine only) can accept. Counting a ball in the barrel is adding to the magazine count and really is not a valid count.
Just now reviewed my notes concerning the airgun used by colonel Bianchini and the ball diameter is described as 5 Linie = 0.432” (10.97mm) which strongly suggests that this would have been a example of the later Nemetz design (as seen in Baker/Currie) and, since this all took place in the 1820s, any airgun in the Vienna arsenal would most likely have been the later improved model.
Remember that the original Girardoni was handmade ("Girardoni - inuenit et fecit") and all reports are that it consistently failed in the field and was impossible to repair. On the other hand, the improved Girandoni (Nemetz design) is reported to be an outstanding airgun, very reliable, and repairable.
It's pretty clear in other Austrian military texts that the armory guys (like colonel Bianchini) did not much like airguns. See Geschichte des K.K. Bombadier-Corps, der K.K. Artillerie-Hauptschule und der K.K. Artillerie-Akademie, 1786-1869 Friedrich Gatti, Albert Edler von Obermayer, 1905 see pages 74, 75, 88.
Obstlt. Vogelhuber and his small team of professional officers, who could hardly meet anyway the duties imposed upon them and had in 1790, taken over command of the main armory office, conducted lengthy trials with an air rifle invented by University adjunct professor Nemetz in Pest.
Meanwhile was Obstlt. Vogelhuber not only by his actual professional duties, but had also still by other affairs intensely been occupied with.
Joseph Nemetz, the already mentioned inventor had indeed lost his job at the University of Pest, but improved his airgun so that, after a single filling of the flask with highly compressed air, it supposedly fired 60 balls successively a distance of 150 paces. Also the weight of the weapon had been reduced by the inventor, but was still 6 pounds and 1 Loth. On July 11, Nemetz, who had become Doctor of Philosophy, a new project presented by which prevented the burning of firing holes and the repair of already burned could be effected more rapidly than before. The proposals of this tireless constructor do not seem to have been completely worthless, for he received a remuneration of 100 ducats and worked thereon for a long time in a laboratory of Bombardier Corps, for which he was paid 2 fl. daily, this was for the price relationship at that time exceptionally high amount to receive.
On 17 May 1799 sent FM Count Kolowrat, on behalf of the Archduke Charles, the drawings provided with a description of a project that originated from a French emigre and former colonel of hussars, Dedouchet*, which was intended to end the danger from charging the Girandoni’schen airgun with the air compression mechanism.** After the Archduke for such proposals always put keen interest in the day, was the submission of an appropriate model to practical tests requested. But the matter was no longer of consequence, because the air-gun Girandoni’sche the initially discovered valuable properties that they had hoped for, it did not possess. The Tyrolean sharpshooters expressed against this weapon an invincible repugnance that one already had to deal with the intention to make them disappear into various depots.