This is a silly question. It asks one to do something that can't be done.
The sentence
- I must go and look for my brothers.
is Intransitive, but Passive can only apply to a Transitive clause.
The Passive rule promotes the direct object to subject, demotes the old subject to an optional object of by, and adds the auxiliary verb be before the past participle form of the main transitive verb. Like this:
- Marie shot my cousin. == Passive => My cousin was shot (by Marie).
But if the clause isn't transitive, there isn't a direct object to promote, and other noun phrases don't usually work.
- Mary slept all day. but not *All day was slept (by Mary).
Sometimes prepositional objects can be passivized, if the preposition is one that makes a transitive verb out of an intransitive one, like look (at) or listen (to)
- We must look at/listen to that again. == Passive => That must be looked at/listened to again.
But that's rarely the case, so most prepositions after verbs don't mark direct objects. And without a direct object, Passive is impossible.
The real solution is to get a new textbook that actually describes English, instead of something like English.
Best Answer
Of your three examples, "I should be allowed to go" is the best. However, "I should be let go" would be more faithful to the original.
One situation when we sometimes need to change imperative verbs to passives is when we are reporting what someone else has said. This is called "reported speech".
For example, if Jack has said, "Let me go", you might later tell someone: