CFW switch can connect online just fine, you only have to make sure you don't have anything related to piracy, and you didn't install any pirated games/apps in the past (tickets remain on the system even after uninstalling the apps).
Also remember that Nintendo can't tell any difference from legal dumps and pirated games. If you have your own dumps installed, they just assume piracy.
You can use Switch Gift Data Manager to inject the Mystery Gifts and claim them in game.