Punch a hole into a RFID card (125khz)
They punch holes in RFID cards all the time. Using a hand or desk hole punch. The smarter companies order them with the holes pre punched. A hand hole punch is fine if you are no where near the coil or IC, otherwise you risk breaking the coil. You may also risk opening the inner layers of the rfid card to the elements if you open it in the wrong place.

The spot you illustrate with green, is just asking for trouble. A millimeter in and you just cut the coil.

Alternative, glue something to the back that would let you attach a rope, like a small piece of fabric or plastic. Or use/make an pouch or card holder.