Ahhh, yeah what I was originally thinking as a solution would have never worked for you. The terms meta is definitely the way to go. This would work for the
card-single.php file too but you would have to create a custom template override file for it to add the functions to list the assigned children by the referenced parent. It would have to be by term ID though, not slug because the relationships are based on the ID. ID/s are safer to use anyway because they will not change, slugs can.
Hopefully this is within @ Kims budget too, hope to hear back soon. I could start working on this after I complete my current custom project and release version 8.1.6 (this project would require code in 8.1.6 anyway).