Program Arduino via UART from ESP8266 SPIFFS
To answer my own question in case someone else ends up here, the library Juraj suggested (https://github.com/ciminaghi/libdfu) looks like it does the job. I also found another project doing something similar (https://github.com/rene-win/esp_avr_programmer). Also the STK500 docs linked to by Majenko (http://ww1.microchip.com/downloads/en/AppNotes/doc2591.pdf) are really useful for understanding the nuts and bolts!