I can't really concur with Dreamweaver's analogy with human development. But Evil Craig F (Ferguson?) is spot on with:
The powers that be must understand that you WILL require entire sprints for re-design of the core module to integrate new work. It's not an optional choice. This will require downtime when the fixes are put into place.
Continuing the building analogy, it's like responding (to the customer's request for a bedroom after seeing the dynamite kitchen): "Sure, but we've got to shut down the kitchen, and tear down some of the existing work to rebuild this into a 2 room module. You won't be able to use the kitchen for a while."
Sure, it's not an exact analogy. More like you're building modular housing that you can crane into position. But you still need time to extend or replace the foundation, and move all the owner's stuff (data) from the one room into the new 2 room module, etc. etc.
Of course, I think most people would concur with Peter Schott that:
I'd much rather start small enough to show something and get feedback as we go. That helps us make course corrections and tackle more important needs as they come up instead of delaying them further.
In my more cynical moments, I feel like I'm one of the workers on the (in)famous Winchester Mystery House...