The big problem is the French Revolutionary/Napoleonic Wars, since Britain could easily take any French settlement there and replace it with their own, leaving the French a small minority. After the Napoleonic Wars, the island is already British. I don't see a substantial population of Frenchmen being established in the years before the French Revolution either--it's just too distant, and France was always bad at populating its colonies with settlers.
Maybe we ignore the distance issues and have France establish a settlement/trading/whaling post in La Tasmania sometime after the Seven Years War in the late 1760s/1770s. Perhaps it gets a few hundred settlers, where in the good environment with good land, they'll have a high natural rate of increase. French soldiers, colonial militia, and other factors (disease, obviously) will let them marginalise the natives so the natives can't unite against them and kick them out. The French send people there as prisoners as well. When the French Revolution hits, have the colony accept royalist refugees from France, plus maybe some worse violence in Haiti causes some more whites to go to Tasmania. Of course, the colony will need to side with the revolutionaries lest it get serious reprisals. During the French Revolutionary Wars and Napoleonic Wars, it resists the British but by 1815 the colony is in British hands. The total white population is maybe 5-6,000 people.
British penal settlements exist, but because of their harshness, gain a negative reputation and few men stay on Tasmania when their sentences are done. Although some British settlement occurs, the island remains largely Francophone. Tasmania accepts federation with Australia due to economic conditions, but ensures that Francophone rights will be respected, explicitly modeling itself on Quebec.
Basically an idea. The hardest part is getting the initial settlement of Tasmania.