I ended up using 4mm-width, 30mm length wood screws, i.e. screws with a conical head and a cross shaped recess (for Philips/Posidriv screwdrivers). So, something like the second from the right in this image:
I used a 3.3mm metal drilling bit - narrower than the screw. Since the metal is thin, and likely not very hard, I was expecting to be able to widen the bore using the screw itself, in a way that would help it fasten itself to the bore. I first drove my drill bit through the plywood frame; when I got to the door, I didn't try to push forward wil a lot of strength immediately, but rather leaned into the drill with mild extra force, and let it take a while to do its work. It took a good few seconds for the bit to go throw the outer metal layer.
I then drove the screw in with a powered screwdriver some of the way, then fastened manually. The fastening required a bit of strength; it compressed the plywood, so that more than the code now appeared sunken; and did seem to "catch" onto the metal - the most important point of the whole exercise.
I did not use any rings.