This is not a GFCI plug, it is an LCDI plug. The copper pigtail 4th wire is a dead giveaway.
The bare wire is a mesh that surrounds the hot and neutral wire. The circuit detects if there is a Leakage Current between the hot and neutral wires and the surrounding mesh, which would indicate a damaged cable. The purpose of this is to prevent a fire hazard. Because the AC pulls a lot of current, a damaged cable can get hot and start an electrical fire. (The internal insulator can be damaged with no visible exterior damage.)
No one really sells replacement LCFD plugs, because (in theory) the design of the cabling requires differently rated current detection circuits.
But you can buy a new LCDI cable from Amazon or maybe a local HVAC shop. Just make sure it's rated for the sufficient wattage. (Almost all will be, but you never know.) You will have to remove the casing from the AC unit to connect the leads internally, which is a bit more hassle than working with a cable. But worth it to prevent a fire!
However, the unit will still work if you tape off the outer mesh wire, and connect the hot, neutral, and ground to a regular GFCI replacement you can get at any hardware store. You will gain shock protection, but lose fire protection, since most of these LCDI plugs don't include GFCI, though some models might. (I wouldn't recommend this, though, don't know if your insurance would cover a fire if it started one. It might be okay as a short-term solution until you can order a new cable if you are burning up and need to cool right away.)
Might be best to remove your defective cable first, so you can visually check if the leads on a replacement will be compatible (sufficient length, compatible fasteners, etc.)