In the end I found a workaround - though I consider it a hack.

I had a look at image_types.bbclass, and was able to create a new bbclass for a new filesystem. Basically it is just mimicking the behavior of the ext4 class, except that rather than an .ext4 file, it just copies over the contents that I need into a directory.

Then I added the new filesystem to local.conf by appending to IMAGE_CLASSES and IMAGE_FSTYPES, and finally made my swupdate recipe depending on that new filesystem being populated by appending to IMAGE_DEPENDS.

I ended up with a new filesystem in tmp/deploy/images that was just a directory containing the files I needed.