[Coda]

I suspect what you'll find is that it won't have a huge amount of effect in boss battles unless the cooldown is way too long. Dashing in combat is done tactically, so having some lag after a dash where you can't dash again might not even be noticed. It's OUT of combat that chain-dashing is most desired, because stages are all about covering ground to get to the boss and everything in between is just obstacles.

Not that it would actually hurt the game all that much, so if it's that much work then don't worry about it (speedrunners can stack dash cooldown reduction if that's what they want). Though 2000ms feels a bit steep (unless that's counting from the moment you press the dash button and not from the end of the dash animation, then it might be okay).

EDIT: I should point out that the musical themed names is mostly for the classic series. It's not nearly so prominent in the X series (the American names of the MMX5 bosses notwithstanding; those weren't their original names) so it feels a little out of place.

EDIT 2: It might be an interesting mechanic to have a single dash NOT consume all of your EAS energy, so you could dash twice in quick succession but then have to wait for EAS to recharge before you can dash again, and longer before you can double-dash again. That would allow for things like wall-bouncing or getting all the way across the boss room in a hurry while not allowing you to always be dashing.