Read just about any top-tier newspaper and you’ll see how they deal with these situations: They use words. “The association,” “the social club,” “the group” — these are all words that exist in the database that is the reader’s mind, unlike AAPWSNBAOSO.
Occasionally, a writer will find herself in a corner when the initialism appears in a quotation. “AAPWSNBAOSO doesn’t want her, either,” said Lance Dulles, president of AAPWSNBAOSO.
The way to deal with this is to set it up beforehand. Find a place to insert, “?.?.?. the association, which members refer to as AAPWSNBAOSO,” and then introduce the quotation.