To make it worse, Cagey, I had specifically warned Tony about masking tape slippage!
What I would recommend, Tony, if you no longer trust tape, is to use your chuck method except make it so that you don't go down until it's touching. Make it so that if you did go all the way down you'd be safe, but that actually you have a mm or two spare. Just to make sure the chuck doesn't spin against the gloss and pull up a big flake and rip it off.
And yeah, loads of wax. If you feel even a tiny bit of resistance, just back the screw out and wax again. They should go in easy.