To follow up, I purchased a Logitech F710 tonight, and what I’ve discovered is that the issue seems to be related to xinput generally. When I switched xinput off for the 710, everything suddenly worked as it should. Previously, both controllers on xinput had another issue I forgot to mention above, which was when I did the controller button configuration process, things went wonky when I got to the triggers.
Pressing the triggers would actuate twice, or sometimes not at all, forcing me to restart the config process. Switching the Logitech off of xinput fixed this issue as well as the lag/freezing problem. Obviously for now, my issue is resolved, strangely by doing the opposite of what the wiki tells me. It would be nice to be able to make xinput work properly, but unless and until that happens, I do at least have a working controller.