Update: My wife decided to make an attempt while I was at work today. After a few transfers similar to my own on Monday, she was finally able to get to a supervisor, and then a "director" at the warranty department.
It seems like the call center staff is trained to consider a broken charging port as similar to a broken screen, as in "almost certainly due to the customer breaking it". Our tablet was not mishandled in the slightest, but I don't think the "front-line" people are allowed to believe that. The "director" did listen, thankfully, and we received an email with a return code and address to mail the tablet in. Hopefully, they'll see that there's something wrong with the port itself, and see for themselves that the pins were not forced or bent at all.
Thanks for the feedback, Daniel and Mike. Mike, were there many issues with the charging port for this model? That's interesting to know. I was definitely bewildered with them telling me that the charging port breaking in 6 months of normal use was in any way "normal".
Will update again upon (hopefully successful) resolution!