much easier to remove the entire fountain assem. from the fridge to work on it.
I usually line the outer edge of the fountain assem. with two layers of painters tape, covering the very dentable thin stainless steel.
Then I use plastic automotive trim removal tools to wedge between the door liner and the fountain.
Starting at the bottom by the drip tray.
Domain and several others I know remove the drip tray, put on gloves that are very grippy(usually coated in a rubber) grip the drip tray area and then simply pull. I'm not brave enough to try this method.
The fountain is held in place with snap in tabs. Once removed from the cavity of the door, you can unplug the connectors and walk away to work on the fountain.