Ok, it looks like I’ve found the origin of the problem by checking if/how it works in Xubuntu (it does). The issue seems to be similar to the one UM had with LightDM before, namely a value for AccountsService that is not set by the settings daemon, except that it’s a different key used this time: Slick (or LightDM since an update?) apparently checks for the org.freedesktop.DisplayManager.AccountsService → BackgroundFile value.
If the user background is set as follows in /var/lib/AccountsService/users/$USER (where $USER is your username), it works for draw-user-backgrounds:
[org.freedesktop.DisplayManager.AccountsService]
BackgroundFile='xxx'
xxx being the path to the background image. For some reason, the path has to be put between quotes, even if there are no special chars in it.
I’ll look into it more thoroughly to fix my old script and automate this.
EDIT: script updated: LightDM doesn't show the user background