I don't believe there is a way to fully disable or remove LiveUpdate (LiveUpdate Engine (LUE) for 12.1), as LiveUpdate is also used in the distribution of content from the SEPM.
You can uncheck the "Use a LiveUpdate Server" in the LiveUpdate policy applying to your clients. LiveUpdate will not run. Edit to add: I just unchecked this in the LiveUpdate policy and I can confirm that this automatically unchecks the "Enable automatic updates" box under Change Settings > Client Management Settings > LiveUpdate tab (if that's what you meant by the GUI box).
Even if this was left checked, if your clients continue to communicate and get updates from the SEPM (default), then LUE will not launch with the message "Symantec Endpoint Protection Manager is available to provide updates, so the scheduled LiveUpdate was skipped." (Edit: This message appears in the SEP client GUI under View Logs > Client Management > View Logs button then System Log.)
Finally, if LiveUpdate did launch and there were no internet, it would just fail and write an error to log.liveupdate.
sandra