As far as I know, the women and girls are NOT told that driving is a Tznius issue, just that it was never “oongenimen” (accepted) or “eingefirt” (instituted/introduced) by them. It seems to be a minhag that they uphold because “my mother din’t drive and her mother didn’t either….”
I like this answer. Chassidim in general try to be/live like it was in the Alter Heim (see: Straimels), and a woman was never the Ba’al Agalah in any of the Chassidishe Maissim (LOL), nor did they drive wagons in any case. The same position today would be the driver of the auto.
If you really want to know, I would suggest asking MSseeker (a Chassidish Lady), who could probably help you.