If one wants to informally exclaim about the excellence of someone, say a barber, using the word badass, should one say "What a badass of a barber!"? Or is "What a badass barber!" more correct? Are both correct enough? Are there other alternatives one should consider?
Definitely "What a badass barber!" The alternative sounds strange.
In informal conversation, "badass" is often used as a adjective, referring to something that has the qualities of a badass. For example: "What a badass librarian!" or "Wow, that sales presentation was so badass!"
You can also use it as a noun: "That fisherman is a badass."
Either usage is grammatically correct; but for the specific sentence you're inquiring about, you should definitely use "What a badass barber."