I am not sure when my project would fit: A six-shooter gun firing small arrows, using 6 PCP cartridge

Think of the 6 chambers like a Brocock cartridge (well, just the idea, not the design). Relatively small amount of air, at relatively high pressure. It would not be a replaceable cartridge, more like 6 chambers that are part of the gun.
Could vaguely look something like this nerf gun at the end, but in machined brass, aluminum and steel:
Note: That's not mine, I found this beautiful paint job on Google image!
I am playing with the numbers for now, I would be grateful if someone could verify that I didn’t goofed! I am learning since a few days about compressed air, and I have little possibilities to verify if I am correct or not (and to my shame I’m not very good with math). I take the example for a single cartridge/air chamber, the 6-shooter grand scheme is for later

Here are my numbers:
- The chamber of non-compressed air would be something like 1” diameter x 3” length. That is 2.36 inch3 if I am correct.
- I plan to compress that volume to 1/8 length (like a syringe with the tip closed), that would result in 0.1 inch3 of air compressed at 338 psi.
- That means the 1” plug would have to support a 266 pound pressure.
- the air exhaust would be 1/4 inch diameter, and that is only 17 pound pressure on it for 338 psi.
- the barrel diameter would be also 1/4 inch diameter, with an equivalent length of 10 inch so that’s 0.49 inch3
Note that if I compress to 1/16" length instead of 1/8", I double the psi, but of course also the pressure on both the plug and the exhaust. That's my power/safety adjustment.
Now the big question:
If those numbers are correct, the air at 338 psi would expand to 4.6x the volume of the barrel ( 2.36 inch3 originally in the chamber that would fill 0.49 inch3 for the barrel). I suppose that would be a good kick. I have no idea how to compute velocity, but the arrow I guess would be something equivalent in mass to a 1/4 inch diameter ball bearing (steel arrow head, plastic tubing and fins.
What do you think? Stupid idea? Could work? Surely not?
