I’ve reviewed other forum responses to this and it doesn’t appear to be possible to remove a Tray Icon without stopping the Service. The app/service needs a “don’t show icon in tray” checkbox. Monitoring software should be this ’sneaky’ but as you state, yours isn’t.
The only other options presented were 3rd. party tools.
Not familiar with them so I can’t recommend anything. You’ll need to investigate them further.

The is a Group Policy item (accessible by running gpedit.msc from the run command) called “Hide the Notification Area”. It is located under User Configuration|Administrative Templates|Start Menu and Taskbar.

Below is a copy paste of what it does…

This setting affects the notification area (previously called the “system tray”) on the taskbar.

Description: The notification area is located at the far right end of the task bar and includes the icons for current notifications and the system clock.

If this setting is enabled, the user?s entire notification area, including the notification icons, is hidden. The taskbar displays only the Start button, taskbar buttons, custom toolbars (if any), and the system clock.

If this setting is disabled or is not configured, the notification area is shown in the user’s taskbar.

Note: Enabling this setting overrides the “Turn off notification area cleanup” setting, because if the notification area is hidden, there is no need to clean up the icons.

click on start the run and type “msconfig” command and then select the startup tab, from there you should be able to uncheck the application that is running at the startup, this will not stop the service.