I bought a used T6 thermostat and installed it to my system using the C Wire and learned it in that way as well. Before learning it in I factory-reset the thermostat as well.
About 1 minute after learning it in, I’m getting “critically low” battery alerts.
Is there anything I can do to stop these alerts without turning off ALL low battery alerts?
You should still insert batteries for back up power. I believe that model uses 3AA.
That is the solution in this case. I would not recommend turning off notifications. Individual statuses for Z-wave devices cannot be disabled in settings.