Replaced the cat with a spare. Initially tried to remove the cat by itself, but the bolts on the front union were having none of it, so I had to remove the cat and the down pipe together and cut through the bolts. Slightly worried that there was no hint of any rattle.
The union plate on the down pipe was very crusty, but seemed to clean up ok.
Put it all back together, using new gaskets either end of the cat. Unfortunately, there is a slight blow from the front union, so I guess that plate isn't as flat as I hoped.
Happily, a short test drive proved that the awful rattling no longer happens. A closer look a the old cat revealed two loose pieces of waddling floating around inside. I have to assume that the core is no longer being held in place properly, and the rattle is the result.
Also managed to confirm that the inner gaiter on the offside drive shaft is split.