Another option is to use an old plumber's trick.
When replacing a washer on a cartridge stem, often the soft rubber
washer compresses too much in the middle before the screw actually
So, around here it was common for plumbers to put the brass bibb screw
through the washer and then use a pair of side cutters to muck up the
screw threads immediately on the back side of the washer. That way, as
soon as the screw was starting to compress the washer, the mucked up
male threads on the bibb screw would enter the female threads of the
spindle and the screw would get significantly tighter. That is
sufficiently tight to hold the washer firmly in place, but without
compressing the middle of the washer all out of shape.
I'm thinking the solution might be to use brass faucet handle screws
instead of the stainless steel ones the faucet probably came with, and
simply use a pair of wire cutters to muck up the threads of the brass
screw only a little. That way, the screw might be a little harder to
turn in and out of the spindle, but it won't turn by itself, and
therefore the faucet handles won't loosen up.
And, that solution sidesteps Higgs Boson's concern about how to remove
the nail polish or Loctite if desired cuz there won't be any nail polish
or Loctite invoved.
Any place that sells fasteners will stock or be able to order brass
screws for you.