It's not totally impossible to make the channel repair invisible but it is difficult. When I've had to do it I've had to use several coats of filler, each requiring drying time and then sanding back. Most trades wouldn't want to have to keep coming back several times. Sometimes it looks great before paint but afterwards any tiny defect becomes obvious.
Can you hide the channels for the wiring behind the mirror?
In my bathroom I got a mirror made with holes in it to match those in the light fitting so the lights are mounted through the mirror into the wall. It wasn't without problems.. I used soft washers under the screw heads to protect the glass but I guess I tightened the screws a shade too much and the mirror cracked down the middle and had to be remade