OK
You do realize that you are very close to something very similar ..
Every time you fire your rifle.
So make your die setup a very short clone of something similar to your rifle action and barrel
A die is in essence a rifles chamber
All one would need is one single valve operated "pop" of air - could even do it from a distance.
We are using compressed air which is a renewable resource rather than components which are consumables
Engineers work out these sorts of calculations all the time
Heavy equipment hydraulics work at these pressures all the time.
I will chat to my machine shop engineering buddy tomorrow
There's likely much more difference between a rifle being used to fireform a case and a valve operated pneumatic system doing it with air than you suggest. Time being the one that leaps out at me. The entire operation is nano-seconds long in the rifle and I very much doubt that could be true in a pneumatic device and therefore the obturation that causes sealing to occur in a rifle chamber is not guaranteed to work the same. It may not work at all at practically achievable pressures and may require a different method of sealing.
I don't get your point about air being 'renewable'. Air just is. Unless you were generating the considerable amount of energy required for a contraption like this to operate from the sun or similar there'd be nothing more or less renewable about it.
Failures in hydraulic and pneumatic systems are barely comparable in their potential for destruction. There's a reason that compressor tanks are often tested with water pressure and it's not because its easier than doing it with air.
Google for some info on catastrophic failures of pneumatic system components for some eye opening viewing. The first such thing I experienced was a compressor that failed behind a nearby mechanics workshop. People reported hearing the bang from well over a kilometre away and two exterior walls of the structure were flattened. One of the guys in the shop was killed and the other was as good as deaf afterwards.