312.2 says that in damp or wet locations cabinets, cutout boxes, and meter socket enclosures shall be placed and equipped to prevent moisture or water from entering and accumulating within the cabinet, with no mention of a requirement that the product be listed.
However, any equipment you use (listed or not) has to be approved by the inspector to be legal (110.2), and I bet that thing is listed for something, but probably not listed for weatherproofing electrical equipment, so it wouldn't comply with 110.3(B) because it isn't installed in a way that meets the manufacturer's instructions. It's really up to the inspector - which in this case is ICE - so he has very firm footing to reject this installation.
Listing and labeling is intended to be a guide to help inspectors be more comfortable with the products they are approving under 110.2. Therefore, if a product is required to be listed under the code, it needs to be listed and comply with 110.3(B). In my opinion, if a product is used in a way it wasn't intended to be used, and what they are using it for isn't required to be listed, but the product is listed for something else, as in this case, it is up to the inspector to decide if they will approve the installation.
I would tell the electrician to effectively bond that cover somehow, caulk the cover to the enclosure to waterproof it, replace the screw with a roofing screw with a neoprene washer to waterproof the screw hole, and call it good. If they don't want to do that, they can install something intended for the purpose. Odds are very good they will buy a product intended for that use, but if they don't I would still be able to approve the installation.
I like to give people choices...