I decided to try connecting it to a PC using the Cyberpower PowerPanel Personal software to see if I could get any response from the UPS that way. I had to dig up an old USB A/B (printer) cable to connect it, but once hooked up, the UPS was detected, and indicated that everything was OK... But even more strange, when I connected it up, the panel and buttons started working again. I unplugged the USB cable from the PC, and toggled the display button and it was working normally after that, and the power button is also illuminated again. So it's odd, but I thought I'd post it here in case it may help somebody with a similar issue.
While I had it connected up to the PC, I also updated the firmware and ran the self-test, which passed.