Latest thoughts after looking at @marcmerlin’s shirt wiring:
This needs to support more than just the controller to panel connection, it needs to connect multiple panels together.
A circuit like this, with SMT female 0.1" connector (ala SmartLED Shield V4) and FPC connector on the same side, could attach to existing HUB75 connectors without any (de)soldering, just a little trimming of the plastic shroud. Height would be 7mm plus PCB thickness, and could be made out of flex PCB. Use two plus a FPC cable to connect two panels.
If the connection to the controller needs to be thin as well, make the adapter with SMT 0.1" male pins instead of female. That could be inserted into the controller’s female socket (e.g. SmartLED Shield V4) or the pins could be soldered to the controller PCB making it even thinner.