System transfers can be done with just the bottom screen.
If you are attached to your streetpass stuff, definitely just buy a new console instead. Nintendo will probably just replace the system with damage like that, and they don't transfer the streetpass data in my experience, even when a transfer is possible.
With the Switch coming soon, might be a good time to move to a smaller 3DS model anyway, eh?
Shame about the damage, though. And to a special edition console, too.