There is a minor counterfactual, and more probably a major...
Louisiana, as we know, was transfered to Spain--officially--to compensate the Iberians for the loss of Florida in the Seven Years War. Unofficially, it was to stop the British from controlling the Mississippi Valley. Under the Bourbon Family Compact, there was a feeling in the court of Louis XV that the colony could be returned from his first cousin if needed.
However, owning Louisiana would still cost the French crown vast sums of money, with little return. The Spanish could justify the colony as a buffer for Mexico. The French had no military/financial justification. It was a constant money loser during the French colonial period. Remember, New Orleans did not become profitable as a port for the Spanish until well into the 1790s.
In theroy, the Spainsh could have returned the colony to France sometime during the American Revolution, when New Orleans was the main entrepot for weapons sumuggling to Continental Forces on the Trans- Appalachian Front. That's the minor counterfactual, and it would require France to surrender one of their profitable colonies. They didn't have much to give, and certainly were not going to surrender Saint Dominque, which provided 1/4 of the national reveunes (thanks to stratospheric sugar imports prices in Europe).
The non-transfer to Spain, after the Seven Years war, would have only come from Louis XV accepting the appeal of a delegation of New Orleanians, led by the founder of the city Bienville, not to transfer to Spainsh control. (They tried; he said no; they consequently planned a rebellion against Spain.) By the time the Revolt of 1768 occurs, and "Bloody" O'Reilly puts down the First American Revolution (as it is often called in the Crescent City), it is too late. The Spanish had to sail nearly every soldier in the Indies. They won't just give up then. Hidalgo pride, even if their commander was an Irishman from Cork.
The only other point of departure would be serious butterflies from a less successful British triumph in the Seven Years War. That would create a far different world.
As for your story premise, while it sounds entertaining, I have to point out two facts. First, New Orleans is not named after the city, but the Duc d'Orleans, Regent for Louis XV. (It was something of an inside joke by Bienville, hence why New Orleans is the only French city that has a feminine gender--Nouvelle Orleans. If you know anything about the Regent, you understand the joke.)
There is no political reason to name it New Bordeaux. And, New Orleanians wouldn't accept a change other than, say, New Paris.
As for the Casquette Girls, none actually came to New Orleans, despite popular myth. They went to Biloxi, prior to the founding of the city in 1719. And, the program was long over by your timeline. It didn't survive the first proprietorship, much less John Law's better known debacle, The Company of the Mississippi.
Plenty of women came to New Orleans, so you can use the topic. Just not that name.