My sympathies - the tip of the iceberg does suggest a daunting mass and now after a year there is a sense of wanting to move expeditiously.... You wouldn't happen to be in a state with inheritance tax would you?
It seems to me that if the family wants to avoid auction and publicity, then for at least a minimum of due diligence, the family should really have two or three offers, based on in-person inspection, rather than just one.
You also have several reactions in hand already proving the wisdom of removing in advance known or even suspected fakes. Pull those you can and set 'em aside. If the family wants to get rid of those later to collectors of forgeries that can be on them, as long as they know that subset of material is bad or suspect.
Wouldn't it be possible to have an auction house run a no-name (anonymous) or pseudonym type auction, thereby keeping any publicity out of the picture? I think I've seen auction houses publish catalogues with pseudonyms used instead of true names (Cherrystone pops to mind)...assuming those stamps not forgeries remaining would hold enough value for an auction house to accept.