The AT firmware should work with all ESP32-C3 products.
Firmware link:
https://docs.espressif.com/projects/espat/en/latest/esp32c3/AT_Binary_Lists/ESP32-C3_AT_binaries.html

Upload method:https://docs.espressif.com/projects/espat/en/latest/esp32c3/Get_Started/Downloading_guide.html