My brief researches only bring up the word "auntlike" to render the feminine equivalent of avuncular. Surely, though, and given the etymology of "aunt" [ < Latin amita -father's sister, old feminine past participle of amāre to love, i.e., beloved ], there must be a more lyrical word to hand.
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Deriving from your own explanation in the OP, the natural choice would be amicular.
I do not seem to find any dictionary entries. Need to see why.
Contemporary Pragmatism - Google Books Result
The correct answer, courtesy of the Oxford English Dictionary, is the word materteral, whose entry I give here in toto:
The proper citation for that entry is:
While their lyrical nature is subjective, dictionaries list both aunt-like as well as auntly as suitable adjectives.