I had called both auction houses the first business day after I posted my original
complaint dilemma--May 3rd 2019.
The larger firm (with whom I have had previous successful bids in the past, amounting to several thousand dollars worth over the past year) responded, with a bit of disbelief at first, but promised a resolution earlier this week.
I have just received their response today, Friday, by email (my invoice):

As I suspected, the bill reflects items I won, but have not yet received. The first item, and the last item, were in neither of the two untracked, USPS-sent catalogues they shipped me in the last two months. The FED-EX sent catalogue I received two days ago, which I signed for, contained only their most recent auction catalogue.
As you can see by the date of the email and the invoice (today's date) it took them just under 90 (ninety) days after the end of the auction (early February closing date) to finally bill me.
And I am missing half the value (the first and last times) Am I to assume they might be holding them waiting for payment? Or more likely they might insist they mailed them, but I have not received them.
Three hundred dollars, now, I am invoiced for items I have NOT received. This is precisely why I posted my bizarre experience with their horrible delivery methods (all items were tucked inside two different USPS-sent auction catalogues which I didn't even NOTICE until last week!)
I wasn't able to check my email until after they closed today so now I will have to wait until Monday at the earliest to get some answers.
I'm still quite upset with specifically their shipping methods! And it appears I will be likely suspected of feigning not receiving items that were sent to me. Unless of course, yet another auction catalogue arrives, with pretty US-discounted postage, with the two missing items tucked inside the auction pages, which seem to be standard operating procedure these days for $595.00 shipments of auction lots.
There is no mention of my EXTREME unhappiness with their shipping methods; the ten dollar shipping fee is not that much less than a standard Fed-Ex shipping envelope which SHOULD have been utilized, as the shipment(s) would have been tracked and required a signature.
Comments? Recourse? I'm very frustrated here....
Jay