Sorry to burst your bubble, but it's unrelated.
The outdoor cooking appliance is called a "barbecue"*; which Aussies simply shortened over time to "barbie". The word even featured in a famous Tourism Australia TV ad some decades ago featuring one Paul Hogan. The name of the girls' fashion doll is merely a coincidence.
Wade.
* And I *know* you USians know that. :-)
"Ah -- I take it the doorbell doesn't work?"