Cut it out
Since you’ve already got the X-acto knife out, there’s another physically aggressive method for slaughtering RFID tags. If you look at a tag, the chip itself will be a small rectangle, usually near the center of a coil of wire (the antenna). The connection points between the chip and the antenna should be fairly obvious. Cut those connections and the chip will be unable to broadcast or receive.
If you want to play it extra safe, cut the chip entirely out of the tag. Flush it down the toilet to show your contempt.