How to prepare my floor to install engineered hardwood?
After calling around, I found a shop in my area that sells plywood underlayment that is 5mm thick, which is close enough to 3/16" for me to use to raise my kitchen floor.

As for installing engineered hardwood on top of existing solid hardwood, I've been advised it's okay when nailing since the material just needs to be void free and substantial enough to take the nails.

If I were floating or gluing my flooring, but not nailing, I could raise the kitchen area with self-leveling compound, but no idea if that'd be any cheaper, easier to do or overall better. The underlayment plywood I found is $14 for a 4' x 8' sheet, so it's cheap, easy to fix to the floor, and should work well for nailing, floating or gluing flooring on top of. Seems like the plywood wins in all areas.