The board looks like an Arduino Nano, or at least a passable clone of the board.
The Arduino IDE has no facility to read the flashed contents of a chip that I am aware of. The Arduino IDE normally uses the “avrdude” program to write the sketch to flash on the chip, and that program has the ability to read it back, but you would need to run it from the command line with the appropriate parameters.
Refer to the following thread from the Arduino forum. The commands shown there are for Windows, but should only require changing file paths and port name to work on Mac OS.
Great advice from Robin to pull the script from your current board. Also, what are the dimensions of the ring, I’m assuming that it’s a neopixel system like that which I’ve linked below, as we may be able to find another to be able to make a replica of the part and modify the script on the new one without having to modify the original.