vintagestg,
Horrors, don't drill anything! My bird is in storage still and will be out in the next few days, so I can't say for certain (its been years since I restored the car and had anything apart).
Typically these units are held in the dash with a 2 small nuts and lock washers on the back of the dash. I believe they are 3/8's inch nuts. Even if you don't stand on your head upside down, you'll be able to feel them with your hand up behind the dash. Its awkward, but not impossible.
Once they are off, you'll be able to pull the entire unit straight out front of the dash 5 or 6 inches or so. You'll then be able to undo the various cables and their fasteners.
Take pictures of the cable placement, and lable with masking tape the cables and their locations on the heater control BEFORE you disconnect them . The control unit will come off in your hand then.
You'll need small allan keys to remove the slide knobs from the control unit. Look underneath each knob for the lock screw.
At this point the face plate's attachment should become clear. I suspect that the backing plate for it has bend tabs or similar that hold it around its edges. (Sorry, its been too many years.) But don't drill it and use screws. Glue it, if you must, or use double sided 3M sticky tape if the mounting is not obvious.
Good time to spray paint gloss black those control knobs if they need it. Good Luck!