Ben, I am pleased to report that as of an hour ago, the problem is finally solved and my replacement battery has been moved to "shipped" status. After continuing to wait with the weekend looming and still no sign of the promised link from the resolutions department so that I could place an order for a replacement battery, I decided to just order it outright and ask them to just reimburse shipping as well as the battery itself. At the last minute I used the sales chat because I had a question about the shipping options, and unlike the support communications, that was answered within a minute or two.
Payge answered my question quickly and clearly, but then went above and beyond. When I thanked her and briefly explained the situation I was dealing with, she took the initiative to ask me to wait while she investigated. She personally contacted the resolutions department and got me the link I had been waiting for them to send me so that I could place the order for the replacement the correct way, waited while I placed the order, and then said, "I will chat our shipping department and see if we can get this pushed out." Less than 2 hours later my battery changed to "shipped" status.
She really went out of her way to get things finally moving, even though I know she works in sales rather than support -- contacting the other departments for me was definitely above and beyond her normal job, I'm sure. She took me from "stuck in limbo" to "shipped" in under 2 hours and was 1000% the most helpful person I had spoken to in a month.
Hopefully the battery will get here quickly, the return of the defective battery will also go quickly, and the reimbursement process will not take too long.