@Robin *********
You can indeed hold a work permit on a Non-O extension based on supporting children.
I would say go to the Cham Churi Square One Stop Immigration Service Center (which handles BOI visas, extensions and work permits) and ask them yourself.
That's where your work permit would come from anyway.
I think your employer isn't wanting you to hold your own extension because then if you lost your job you wouldn't lose your current extension seeing as it's not based on employment (meaning they lose some "control" over you)..