He's right with the 4 BA.
Your wanting to thread the thin wall of a cylinder and used good thinking in going for fine thread in attempt to try and get maybe 3 or 4 threads into the slim wall which I've also done.
You can find M4 fine taps on Amazon for a few quid.
I've also run an M4 fine die down regular M4screws with decent results.
Both metric fine die and taps available on Amazon for hardly any outlay.