Again, 5 cf is 37 gallons

. You could potentially lug that around, and it might hold enough air, but that is teetering on the edge of impractical.
I imagine you are thinking of piping the air in to the intake before the turbo (because if you were to do it after, you would likely stall your compressor, no?).
I hadn't got as far as figuring that out, but I don't see how stalling the compressor is part of the equation; the compressor is in the system just to charge up the holding tank, not to try and keep up with the flow itself.
Think of it like filling up a cistern by ladling cups of water into it, so easy a child (in this case, a tiny little compressor) could do it given enough time, and yet you can dump the whole thing... no wait, better analogy, toilet tank. Slow fill, fast flush.
Your system is going to have to be contained in some way so that your charge of compressed air does not find its way out the intake. Some type of check-valve might do the trick.
Systems exist which use both a turbocharger and a supercharger. Where does the supercharger add its boost, in those systems? That's where this boost would go. It is, conceptually, an electric supercharger, just one that gradually builds up the pressure over a period of minutes rather than being capable of generating the full required flow rate in real time.
Even then, I'm afraid that 250 cfm is going to be less flow than your engine already needs. According to this
calculator, even 350 cfm is only going to yield about 225 crank hp. So you're gonna need an even bigger tank if you want to increase flow (and power) above conventionally achievable levels.
Given that we already have a 138hp engine (or more, in the case of those of us with mods) we're talking about taking that to 363 hp with your hypothetical 225 hp additional capacity. I wasn't thinking of going that high though.
Remember the idea here isn't to substitute tank air for normal boost, it is to add tank boost to that already being produced. So say we produce an extra 50 hp worth, that would require only 32 CFM.
It's a neat idea, but I really think that storage capacity is a serious limitation (and just think of the weight of the tank and compressor).
You might be right, I'm hoping that a reasonably small tank and an unreasonably small compressor would work, the whole idea is to get around the unreasonably large "compressor" (because that's essentially what a supercharger is) by substituting storage.
Another, alternative plan, if the compressor itself becomes unweildy, would be to use a separate compressor, which could be hooked up at the workshop or in the "pit" area at the track. But that really sounds like more trouble than it's worth.
They run fork lifts and even cars on compressed air, surely there's some way to store the amount needed for combustion?
Seems to me that a much more prudent alternative would be to get an oversized turbo and feed it with a small bottle of N20 for spooling purposes only.
Might be... probably is, since that's the sort of thing people are doing, while this compressed air idea seems to be a new one. But the dream here is to substitute a few pounds of storage and a cheap compressor for thousands of dollars worth of upgrades that are only used a few seconds a day.