I just did a calculation and if I ignore the correction needed for 300 bar air and assume it behaves as an ideal gas then I too get an answer of 8 fills. It's wrong.
If you want to read up about Van Der Waal then do a google search. Basically what happens is this; as you compress air, you are really cramming more and more molecules of air into a confined space. Initially if you double the number of molecules then you double the pressure. Air behaves like this (approximately) up to a pressure of about 232 bar (what a surprise to see that number coming up!) because up till then there is "room" between the molecules to get more in; the new ones just take up the free space.
Above 232 bar it all changes. There is less and less room to get more molecules in. They start to rub shoulders and press against each other, and the additional pressure on the gauge comes partly from their resistance to accommodate any more chums, not from additional mass of air going in
Now it is the mass of air in the bottle that does the work, when you fill your gun, and similarly in the gun when you fire the shot. As there aren't as many molecules in the bottle as 300 bar might lead you to expect, 300 bar can't do the amount of work that you would expect either.
Apologies to the true engineers and scientists for this Mickey Mouse explanantion.