Three way circuit powering a GFCI outlet
You can switch a GFCI outlet like you describe. Obviously the green LED indicator is powered from the power coming to the outlet so it stands tor reason that it would go off when the power to the outlet is switched off.
It is possible that some loads that you plug into an outlet like this could cause nuisance tripping of the GFCI when the switch is turned on and off quickly. This would only be likely to happen with devices that do not have properly balanced front end AC EMI filters that are tied into the safety ground.