Wowwww, this is strange.
If everything happened as you say it did (and I don’t have reason to believe that it didn’t) then it doesn’t seem to me that you did anything wrong. And it certainly doesn’t seem to me that you rejected that particular pile of clothes due to racist intent on your part. As you said, you did, after all, accept some of her clothes.
I haven’t read too far in the thread, but I did see that “playing the race card” was mentioned. (God, how I hate that phrase!) I don’t think it’s quite that simple. IANA psychologist/psychiatrist, but it seems to me that there could be some kind of chemical/mental imbalance going on. Not saying that this is the case; only that I wouldn’t discount it as a possibility. Again, like you said, you did accept some of her clothes.
And the thing with the mother…sweet baby Jesus. I understand a parent’s instinct to protect his/her child, and I understand that this white woman being the mother of a black child likely gives her some kind insight into under-the-surface white racism that disimilarly situated whites don’t seem to have (or many, for that matter, aren’t particularly interested in acquiring), but…dang, even racially enlightened white folks can get it wrong sometimes. I’ve certainly seen it happen.
Anyway, my first instinct is to tell you not to apologize since, on the merits alone, there’s no evidence that you acted in a racially discriminatory manner. On the other hand, though, I can sometimes be a softy, and it does seem as if this girl has some issue going on that miiiiiight be somewhat ameliorated by some gesture on your part. So, maybe you can call her up and tell her that you’re not a racist (okay, not a capital “R” racist, anyway, as I’ve mentioned in another thread), reiterate just why you rejected the stuff that you rejected, and tell her you’re sorry that she (obviously) misinterpreted something that you may have done (she may not, after all, be aware of the customer/consigner/phone priority that you mentioned, and she may have been put off by what she imagined to be you interrupting your transaction with her to tend to someone else). That’s it. No more, no less, then let the cards fall where they may.
If you do this on the phone, you might want to have another party listen in (someone objective would be nice) so that you can’t later be accused (not that that would necessarily happen) of doing something else improper. Better yet, perhaps you can invite both her and her mother in for a meeting with you and a (preferably objective) third (well, fourth) party and have the conversation then.
Good luck!