I don't consider the word "inquire" formal though it may be a tad more formal than saying "ask about." But no one would blink an eye if you "inqured" about something. For example, a sign posted in a U.S. store window advertising that the store is hiring will often include the phrase "inquire within."
Neither do I think an "investigation" automatically comes to mind when an American hears the word inquiry...unless it is pronounced "in-qwa-ree." (Spoken in three syllables instead of four: "in-qui-er-ee.") A good many Americans *would" think it was a gross misspelling if they saw "enquiry."
We do use terms such as "booking a flight" so I think "booking inquiry" would be totally understood. "Booking enquiry?!" Well, expect a lot of email from Americans kindly informing you of the spelling error.
Few, if any, Americans would ever write the word enquire. It is foreign to most of us unless we read international papers, particate in online forums with international participants and the like. Most Americans still think it looks odd when someone writes "whilst."