Final update to my original problem:

Attempt #4
I went ahead and ordered the product a 4th time! Now I know I'm really a fool. I entered all personal information exactly the same as before, except this time I gave my work telephone number rather than my home phone number. That way, I'd hopefully get their fraud check call rather than nobody being home.

That call never came. Instead I just received my shipping confirmation message including FedEx tracking number!!! No idea what's different this time that's allowing it to be processed, but I'm not arguing. The whole process feels almost random.