What requires them to get approval from the FAA for something outside the FAA's jurisdiction?

The FAA claims jurisdiction worldwide. Did you not see the case where Musk launched from India because he could not get reasonable US approval and they pitched a fit?

I recall that but I don’t think that was SpaceX. Some other satellite firm.

I don't think that's true.

Many aspects of SpaceX's rockets are covered by ITAR, which puts strict regulations and controls on the exports of many aspects of SpaceX's vehicles.

I imagine exporting the rocket to a non-US based firm would be a much larger bureaucratic hurdle than getting FAA approvals.

ITAR is largely voluntary. I say this having looked at the regulations because at one time I wanted to produce/was producing things that fell under ITAR. In a lot of situations if you say it is not restricted by ITAR then it is not.

This leaves me (and hopefully you) wondering what ITAR actually still accomplishes. I guess federal contracts dictate the use of the ensuing technology? I still see issues with that depending on how IP rights are assigned.

I guess we're waiting on the next PGP-style test of export control laws.