Actually, GeForce Experience used 300x400... for most cover art. Overcooked's is 628x888 pixels for some reason. Moonlight client themselves should all just adjust to whatever size the cover is, as long as it's close to a portrait aspect ratio.
Sunshine does have a tool to import from IGDB, you're right, but nothing prevents you from using SteamGridDB. If you wanted to request that Sunshine adds SGDB as a tool alongside IGDB, you can create a post for that, I'll tag it accordingly (though I don't know if the Sunshine project will be able to add your request)