The cool/fan/dehumidify/heat mode is indicated with a vertical column of LEDs (EdgeStar AP14001HS), so I can probably stick a photocell at the top, and have a program repeatedly send "mode" while watching the brightness level, to deduce the state and stop on the correct one. The fan speed uses a different column of LEDs, but stopping on "dehumidify" forces the fan to low. The temperature can be set blindly because it stops at the minimum/maximum value.
I haven't actually tried to build anything yet, but it at least seems possible using a Raspberry Pi with 1 analog photocell and 1 IR transmitter.