Thought crossed my mind, can we power the Pico (or other board) from the 'phantom power' provided via camera's mic port?
If so, we could make 'pico-timecode' into a dongle to plug directly in. Once sync'ed it could remain attached to/power by camera.
There is this small variant of a Pico board, which can also run micro-Python....
https://www.pishop.ca/product/tiny-2040-2-mb/