I'd just solder paste and reflow, like a large surface mount device. Challenging to get consistent no doubt, and the alignment would have to be very accurate (or have a few separate boards), but I think doable.

Seems like you could largely automate a workflow for identifying pads in the scan and generating the mirror layout, with simple routing to some kind of standardized interface for the probe lines.

Might be able to find pogo pins that small. That would be my best idea (if they're available).