Quick update in case it helps others —
I decided to book with a company called Gotham Ride, based on a friend’s recommendation. They offer flat-rate black car service with meet-and-greet at JFK.
I’m flying in late next week, so will report back once I’ve tried them — but the booking process was super smooth, got a confirmation right away, and they guarantee the driver will be waiting inside the terminal with a sign.
Website is gothamride.com if anyone’s curious. Will update after the trip!