So, I bought a new external USB drive enclosure and a shiny new 1TB
drive for backups. When I connected the drive to the server, the RCD
tripped. The UPS also failed.

Make a mental note to make sure that everything under /etc is also
backed up, because umm it wasn't this time.

And the cause of the problem? An earth leakage fault in the PSU of the
external drive which only became an issue when the USB lead was
connected to the server (or indeed if the USB lead merely touched the
earthed metal case of the server).