Why can't I use a regular modal verb here?

This question appears to derive from the 'Entry Test' at the beginning of Diana Hopkins with Pauline Cullen, Cambridge Grammar for IELTS, 2007, and the answer from the 'Key' on page 223. You are invited to read Unit 14 for an explanation of the answer. If you do so you will find these among the 'rules' given there:

We use must when the obligation comes from the speaker.
You must invite me to visit you. (the speaker wants this)

When there is an institutional rule or a law have to or need to are more common than must:
You have to get a work permit before you go. (this is a rule)

We do not usually make questions with must and ought to:
What sort of things do you need to know? (not What sort of things must you / ought you to know?)

There is thus, even on the author's terms, absolutely nothing wrong with must I do in this context; "more common" and "usually" are statistical observations, not constitutive linguistic rules.

My personal opinion is that this sort of neo-prescriptivism is a methodologically unsound abuse of scholarship, and that its incorporation in formal tests which profoundly affect students' academic and career prospects is an outrage.

But I don't get a vote on these matters. Mss. Hopkins and Cullen are unquestionably more familiar than I am with the canons of grammaticality embraced by the IELTS writers, and we may probably assume that their book is a reliable guide to the sort of rubbish you need to know to succeed on this examination. I advise you to pay close attention to everything they say and follow it to the letter—until you have passed the examination and are free to follow your own linguistic instincts, which at least in this instance are impeccable.

Answer A would come to American lips more readily than B. (What do I need to do to get a drink around here?). Answer B is a little more fastidious, a little less colloquial, perhaps, but sufficiently idiomatic to find its way into the lyrics (and title) of rapper, Lil' Zane's 2000 recording, What Must I Do?. It is also used in the same sense in most English versions of the Acts of the Apostles (chapter 16, verse 30). There is no reason to believe that answer B is anything other than standard, idiomatic English.