His public career as economist started only in 1844, and ended with his untimely death in 1850. Bastiat had caught tuberculosis, probably during his tours throughout France to promote libertarian ideas, and that illness eventually prevented him from making further speeches (particularly at the legislative assembly to which he was elected in 1848 and 1849) and took his life.
Bastiat definitely embodies the "Harmonic" school of libertarians, who consider utilitarian and natural law arguments as two complementary aspects of a same world.
Bastiat did not take part in the anarchist vs minarchist debate (he arguably died too early for that); he seems to have considered State as something inevitable as far as immediate practical things matter, something that ought to be taken into account as long as it existed. He also explicitly deplored violent revolution as a way to get rid of governments. Finally, his friend Gustave de Molinari did publish his foundational work on free market anarchism in 1849, and Bastiat, knowing that, did declare on his death bed that Molinari was his spiritual heir.
Bastiat is the author of the satirical document best known as the "Candlemakers' petition" which presents itself as a petition from candle-makers to the French government to block out the Sun to prevent its unfair competition with their products.
His stress on the role of consumer demand in initiating economic progress meant that he was seen as a forerunner the Austrian School.