I discovered that pieces like this are best played on a Steinway piano, which has a better action for the fast multi-tapping of notes that goes on. On pianos with a more sluggish action, a lot of those fast notes get lost because the key doesn't bounce back up fast enough to be hit again. (Actually it's not the key, but the hammer inside... but we needn't get into that. :) )
So yeah, if I'm ever gonna really perform this like at an open mic night or something, I'll need to either have a Steinway piano, or an 88-non-weighted-key keyboard. Neither is very easy to come by... :-/ There is a Steinway in the music room (which is how I know they're good for this stuff, plus I heard Billy Joel won't perform on anything but a Steinway, I assume for similar reasons) but it's a bit out of tune and not exactly portable.
Oh well. I need to learn the rest of this piece first anyway.