Can the air conditioner make the battery light come on?

Hello, the battery light, a burning smell and smoke typically occur when the belt is slipping on a binding pulley. As an example, if the air conditioning is turned on and the compressor is binding, the belt continues to be driven by the crankshaft and burns on the frozen air conditioning compressor pulley...cause the alternator to slow or stop also which causes the battery light to come on. Did you check to see if there's even a belt on the pulleys anymore? My guess is that It has broken off.

Additionally - IF the drive belt is still intact - removing the belt may allow engine to crank - assuming the a/c compressor has locked up. Ultimately - in any event *hands-on* / evaluation inspection is going to be required here.