If you get desperate you could use pogo pins - make a PCB with pogo pins in the right positions. You wouldn't insert the card, rather you could lay the card on the pogo pins and secure it with something. Maybe mill a holder, some kind of socket from plastic?
You can get really small pogo pins. It might cost a pretty penny, but since it's aviation-ish, money is probably not an issue.
edit:
I noticed you had this same idea
...but I've also been considering something with pogo pins. I've never worked with them so I don't know what to look for on Digikey.
Dunno if they have them on digikey, you should look directly at the manufacturers catalog. See manufacturers such as feinmetall (
http://www.feinmetall.com/products/contact-probes/ ) or ingun (
http://www.ingun.com/en ). I know that farnell does not stock them since there's so many of them available. I've had experience with both of these manufacturers, both are good, both will sell directly at low quantities - or rather their local distributor will
. Or you can go the ebay route and order a load of crappy probes from China for a fraction of the price
See feinmetall catalog:
http://www.feinmetall.com/fileadmin/Downloadseite_Dateien/Gesamtkatalog_Kontaktstifte_2015_EN.pdf