Balsamic vinegar is allowed only when very aged and with no hidden sugar or starches.
Here's excerpt an older post:
I recall that Elaine said that real balsamic aged a long time (15 or more years) would be legal as long as there are no added illegals, including no added sugar. Grape must is a legal ingredient in vinegar. I also found this post from Marilyn: "...GOOD, EXPENSIVE balsamic is okay. Cheap stuff, with sugar added, is not. 20 years fermented vinegar might be better if you can get it.
Marilyn" from https://groups.yahoo.com/neo/groups/BTVC-SCD/conversations/messages/123218
So if the supplier will give you a letter that confirms the ingredients, it may be fine.