The tape is exactly for that: The Wii U uses a laser to read the switch state but this laser is pretty picky. A small scratch between the switch and the casing you can't even see is enough for the Wii U to think the card is locked. In these cases one just grabs a piece of tape and covers the switch under it: Laser can't pass anymore, Wii U thinks cardn isn't locked. This trick even worked for people who somehow managed to completelly rip the switch off the SD card.
So seeing the OP tried tape already I'm pretty sure it's not a write lock issue.