I don't think a law like that is really a problem. There are already laws on the books similar to it (pawn shop laws and such) where KYC is a big thing to prevent the fencing of stolen goods.
It can be as simple as this:
1. Joe brings in a cat
2. Store takes the cat, but writes down Joes important information (DL #, name, address, etc)
3. If Joe brings a new cat in within some reasonable time period factoring in possibly fixing a used car or something he's reported to the authorities for suspicion of theft.
Exceptions to (3) can be made to people who can present the valid credentials of an auto repair shop and are operating as agents of that shop. Then the shop can be placed in the record book and tracked with different standards.
With this in place you will only be able to fence X number of cats easily where X is the number of shops within some reasonable distance. You could even make this national if you really wanted to prevent transportation over a border.
Sure, you could argue this won't fix anything because shops that are dirty will remain dirty. This is simply solved by having an already existing traffic enforcement body once a year check books. If your books are out of order your business is closed and an investigation is done to see if you're acting as a fence. Same as pawn shops.
There is absolutely no "intrusion" to speak of here. You are in possession of a highly valuable, commonly stolen item. KYC by a company should be a minimum standard. Do you think that requiring a car title and asking for registration, etc when you sell a car to a lot is also an intrusion? I'm afraid to ask you if you even know what fencing actually is. No one is saying you can't cut your own cat off and sell it privately. The goal is to eliminate the easiest possible routes for fencing and make it not only difficult but also expensive criminally to continue.