What do you call 'be-to' constructions and are they acceptable English?

Solution 1:

It's the Be + Infinitive Construction, if you need a name. The construction is not limited to infinitive use; i.e, it's not just to be to + Infinitive. It can also be used in the present or past tense.

• He is to leave tomorrow.
• He was to leave yesterday, but the flight was cancelled.

A bare infinitive in a strange place, like after an auxiliary verb (e.g, be or have), is bound to be an idiom, and usually has a modal meaning:

• He has to leave. (He must leave)
• He is to leave (He will leave ~ He is expected/scheduled to leave)

just like relative infinitives (infinitive clauses modifying nouns) always have a modal meaning:

• He is the man to see about this. (He is the man that one should see)
• He is the man to do the job. (He is the man who should do it)