Knowing the ways to pronounce Claude Frédéric Bastiat is not nearly as important as understanding the concepts that underlie his elegant writing. Have patience with those that follow another pronunciation key.
Bastiat was French so he would have pronounced his name in the French manner, bästyä´ (bast-ya); frequently a very soft 'e' sneaks in between the two syllables, almost as an accent mark on the hyphen. While this might be regarded as the preferred pronunciation, it should be realized that French government tends to disown Bastiat; not teaching him in schools, nor studying him at the universities. When you read Bastiat and then view modern French politics it is easy to see why they choose to avoid him.
On the other hand, Americans tend to pronounce Bastiat as three syllables, and in a very unFrench fashion pronounce the last consonant while putting the accent on a hard middle e. (bast-e-yat). Of course modern American politics is no better than French politics; we must make room for this pronunciation because America possess more of the logical sort of student that embrace Bastiat's ideas.
I can not speak for Frederic Bastiat; but for myself I would rather have my ideas discussed openly, than have my name pronounced with the proper inflection. Bastiat did say: "The worst thing that can happen to a good cause is not to be skillfully attacked, but to be ineptly defended." - C. F. Bastiat
Bastiat was an able defender of his principles.
Much more important than pronunciation is the elegance with which Bastiat stated his good cause. He could in one clear sentence sum up ideas others would dawdle over for hours.
"Everyone wants to live at the expense of the state. They forget that the state wants to live at the expense of everyone." - Bastiat
A single paragraph by Bastiat could relate arguments it would take others a book to outline.
"Away with the whims of governmental administrators, their socialized projects, their centralization, their tariffs, their government schools, their state religions, their free credit, their bank monopolies, their regulations, their restrictions, their equalization by taxation, and their pious moralizations! And now that the legislators and do-gooders have so futilely inflicted so many systems upon society, may they finally end where they should have begun: May they reject all systems, and try liberty; for liberty is an acknowledgment of faith in God and His works." - Claude Frederic Bastiat
If you want to understand the concentrated effect of Bastiat you can read a short broadside like The Candlemaker's Petition. In a page or two he establishes an entertaining argument that exposes the reasoning of those seeking government protections.
And if you want an education in reality; Bastiat's essays The Law and What Is Seen and What Is Not Seen will provide you with valuable insights for everything from starting your own business to understanding Hollywood politics.
Perhaps you want even more. There is available a two volume set of Bastiat's writings translated into English (video discussion uses American pronunciation of Bastiat). The Bastiat Collection has enough economic theory, entertaining insights, and clear thinking to help you refute arrogant supporters of bureaucratic plunder. Consider reading this your senior project in the principles of social interaction.
Go ahead and pronounce Bastiat the way those around you pronounce Bastiat. Just be sure to understand what genius that name represents. Also be open to others that pronounce his name differently, but have the wisdom to seek his wisdom.