A patch should work. Definitely don't waste your time on a plug. The hole is too big for that. Do tire places still vulcanize tires? I've been out of the loop for tire repair lol.
Shop repaired a nearly new tire on my Charger with a very similar hole... they used a plug w/a patch on it like the one pictured.
It never gave me any issues, including several long road trips.
My only concern would be if it caused any damage to the steel belts within the tire. If you start to hear a thump thump thump from that tire then it's game over