I uninstalled BM, and as soon as I did so, our RMM agent showed the server as offline. I was able to get it to show as online again by manually restarting the service, but now I'm having applications telling me that this server "does not have internet" even though it does. Might BM have uninstalled/disabled/changed something during its uninstall process that is causing these issues?
What you describe is not supposed to be happening. Bandwidth Manager does install a lightweight filter driver for traffic capture, but it should have been safely removed by the uninstaller. Something must have happened during uninstallation, for example some other software interfered with the normal process.
Please check in the adapter properties in Windows settings if the filter is still installed. If so, you can remove it manually, as shown below. Otherwise, a simple reboot may help.