I'd try some spray Contact Cleaner. It's an alcohol based cleaner that desolves grease and then evaporates. Many times it contains a little bit of lubricant, too.
It usually comes with a spray tube, like WD-40 does, so you could direct it directly into the tiny space between the switch and it's casing. So, you might not even need to get behind the switch to spray it into the switch.
It's certainly worth a try and you can find it at any Home Depot or electronics supply/parts store.
And of course, on Amazon:
https://www.amazon.com/CRC-05103-Ele.../dp/B000BXOGNI