Wash away filament! That's the real solution for 2-headed FDM printers. For your situation, I'd simply infill the die and call it a day. Offer both up on thingiverse.

I've created some surgical level supports for things like this before. Maybe you can include an internal structure that can preserve the hollowness of the die?

Small feature do have some trouble resolving simply due to flex. A 200um print may come out better than 100um. My thinking is that maybe the nozzle is imparting some damage during moves. Z-hops are always a consideration for little parts like this. Sometimes they help, sometimes not.

Cooling towers? (pause to allow the current layer to solidify sufficiently?)