2 button remote controlled by single input

You are going to have to build something to mechanically push the buttons as you say you do not want to mod the remote.
Using an Arduino with a plug on connector shield that has connectors for the servo to pluh in is quite easy, and available ready made on Ebay. Or get a proto shield and just add the header for the servo and a couple of bits to sense the power state. Very few parts required.
Look for Arduino Uno and Arduino Sensor Shield.
The servo will just plug in.