In the GoG Galaxy settings panel under the Notifications page I have notifications for Wishllift alerts, sales, set to neither desktop or overlay, but off. But I'm constantly getting a flashing icon for GoG Galaxy on my system tray due to new wishlist notifications.

I'd like to be able to stop that. But changing the appropriate settings for the client don't have an affect.

How about try adjusting these settings on the GOG website itself (login to the website), then restart the client and see if it works? This sounds like a bug. You should definitely report it.