When the ignition is in START, the radio will always be off.
This is what I'm thinking based on your statements that:
1. Lights work in ON until the headlights come on.
2. Lights work in ACC, regardless of headlight switch position.
Correct?
I second going to an electrician. Without being able to physically look at the car, I have no idea what they did to the wiring, but something is off. We won't be able to help you here because we have no idea how the Bluetooth was installed.
If you want to try it yourself, I'd start by pulling the radio and looking at wires to see what is tapped where. I want to say that it has to do with the headlight signal wire that goes into the radio (it acts as a signal to tell the radio when the headlight is on or not, and on some stereos handles dimming as well), so they might be lugging off it, for whatever reason they did (probably confused it with the power wire).