The gerund clause being chosen is indeed in the passive voice. Its Subject (the Patient, the person or persons who are chosen) is inferred from the context.
But this clause is not an independent clause—it cannot stand by itself as a sentence. Only clauses with finite verbs can be independent. (Finite verbs are those which are inflected for tense, and sometimes person and number. Infinitives, past/passive participles, and gerund-participles are nonfinite.)
The main clause does have a finite verb and it can stand by itself as a sentence: X is not enough. The gerund clause acts as the Subject of the main clause, X. What is not enough? Being chosen is not enough.
The main clause is in the active voice. Copulative BE does not take objects, only predicate complements; it cannot be cast in the passive voice, even though as an auxiliary it is a necessary component of the passive construction.
As you conjecture, however is a kind of "parenthetical". It is not a constituent of the sentence, a piece which is necessary to "constitute" the sentence, but an adjunct—something added.
In some contexts enough acts unambiguously as a "fused determiner", but here it is very difficult to detect a head with which enough has been fused. I'd see it as an adjectival. But whatever else it may be, it is certainly a predicative complement of is, which is all that is relevant to your question.