Just how immediate is "immediate"? And how much budget is there to achieve it?
I'm not sure you realize just how much is "possible" to attain here, given any price whatsoever...

Put another way, what's the
slowest acceleration that is still satisfactory for the application?
Using a "toy motor" sounds like it doesn't need to be all that fast at all, perhaps seconds? Which the batteries will more than suffice to supply, no additional storage needed.
Any interest in regenerative braking, to extend battery charge? (Added complexity of course, but might come for free with the right motor driver.)
Is the wheel unloaded? How about bearings, and their losses? Is it spinning in air, what about windage loss? Anything touching it? (And if nothing is touching it, does it really need to spin at all? Might there be another approach entirely that you haven't considered?)
Tim