Translation:Un mois complet ?
Does "complet" always come last?
It is not a BANGS adjective so it's placed after the noun.
Un mois entière = not right?
mois is masc., entier.
Would it sound strange to put entièrement instead of entier?
Yes, because entièrement is an adverb. You're describing a noun so you need an adjective. In English: "an entirely month" would sound odd for the same reason.