"Must have been is correct". You can't use a past form after a modal verb like must, can, should and the like. Since you can't change the form of the modal, you change this of the infinitive (using a perfect infinitive) to give it a kind of "past shift"
He should come (Perhaps he will)
He should have come (But he didn't)
And about "professional career", it doesn't sound awkward to me; I think it is a usual collocation.
I hope this is useful
See (or read from) you