What I don't understand is why there is not a system in place that offers a "BetBooklet" pass or something (not necessarily a *card*, per se) that the Bettor applies for as a condition of betting with that Bookie or group of Bookies.
This pass would allow all bets made by that person to be ID'ed by a random number. It would be up to the Bookies to hold (and protect) the personal information of the Bettor (as it is now), and the only data that needs to be shared to the Sporting Bodies(tm) is the pass ID number. Then, if there is a problem, the Sporting Body(tm) makes a formal request to the Bookie, according to Data Protection principles.
What about multiple Bookies, you ask? If needs be, a separate service could be installed that does this, WITHOUT holding the person's information: when a Bookie makes an application to the Service for a new "BetBooklet" pass, a standard hash is made of the person's real (formal) name from a standardized form of ID and the ID of the document used. In the case of USA, a driver's license number or (note, not either - the same document type must be used across the board per country.) Social Security Number (God knows I won't be able to collect social security on mine, even if I live that long). If you don't have that kind of ID, you won't be able to bet anonymously, but you also probably would not be betting those kinds of amounts of cash that would want tracking, either.
This way, the anonymous service would be able to vouch for the authenticity of the Bettor based on the Bookie's held data (which would be the only copy), satisfying the Sporting Bodies(tm) tracking requirements, and all without violating personal data privacy. Heck, the service could even be in the US and not violate UK citizen's privacy at all (unlike the PNR fiasco) because the Service *never* knows what the personal data actually is!
In fact, you may make inquiries to my email address if you would like such a system in place (a man's gotta make a living, ya know). I could probably get it done in a fortnight, at that.