Hhmm, interesting thought Free.
I have a book called Poor Law Union Records which lists all the places covered by registration districts. For Barnsley there is no mention of Barnsley Union, just Barnsley.
Looking on freebmd gives the same, no specific mention of the Union.
It may well be that the GRO are being specific when a child is born in a workhouse. I guess one way to check would be to check the census record to see where the child is living, especially if the child was born close to a census year.