I drilled out the 2mm pilot/test hole to 5mm, and tapped to 1/4" x 20 TPI.
I fixed a 1/4" pan head bolt to the end of a dowel with blu-tack, and inserted it all the way up the tube, and managed to run it partially through.
I then grabbed the protruding end of the bolt in a soft jawed vice, and tightened up.
I think the pan-head was slightly too large, because the bolt didn't come through as far as I though it would; I may yet run it back out and file the head down a little.
But - in principle - I have a method of fixing a 1/4" stub to the tripod, and thus a way of replacing the broken head.