I'm not sure about the rechargeable aspect, but for programmability you could use a small IR receiver placed in the center of the LED array. You really only need a one way data transmission and something like Bluetooth is overkill.
Could set the earrings on a phone screen and transfer the patterns over. Sounds like a fun project.
Here's a (tiny) demo of this for my PCB business card project from years ago [1]. If IIRC this proof of concept was as simple as using a phototransitor on a GPIO connected to the UART peripheral with a very low baud rate.
[1]: https://www.matt.egan.me/entry/electronic-business-card-pt3#...
Concerts with audience-worn LED bracelets regularly shoot out LED floodlights to the IR sensors on each wrist. Being able to do something similar on people’s ears could be cool.