Hmm I guess adding another parameter for synthetic full backups would be the solution.
For your specific problem it would be better to have image backups with virtual clients though. Your database data gets regularily added and removed from the backup images wasting space.
What would work currently is excluding it always and backing it up as files.
This is what i do, i.e always excluding databases and running a SQL task on week days to backup db on a NAS share on the remote site (clients are internet clients).
But the must would be to have at least a “full” image including SQL databases per week on the main site (headquarter).