Now whether you get it populated or not..
https://github.com/Nicholas-L-Johnson/flip-card/blob/main/ki...
That is based on the price for low quantity (1-500 pieces) though; if you were to build more than one board you would buy more LEDs, pushing the per-LED price (way) down. You can get 4,000 LEDs for $30.'
Edit: here is the LED in question, from the BOM [1].
Are there services that you can send the BOM and board files too, and not only do you get the PC board back, but it's populated? Will they do that for a onesy-twosy thing like this?
Then all you have to do is supply the battery and download the firmware (however that is done).
There's definitely an entire business available for expensive trade show merchandise, including electronic business cards. People routinely give away shirts and other merchandise that cost far more than $10..