I was always able to re-order the categories, but my problem was that they wouldn’t save. When I reload the page, they’d revert to the previous position.
Regardless, I tried testing this on a different environment and couldn’t reproduce the problem. Even when reverting to a generic theme and disabling all other plugins, I couldn’t eliminate this problem in my current environment. There seems to be some problem within the site, somewhere that I can’t pinpoint.
I’ll consider this resolved since I can’t reproduce this error elsewhere. Thanks for your help!