And they will do a shitty job of it
If your check doesn't say "Sportsbook, Inc" or something like that, I am sure they will cash it. If you send your money to an agent, not the book, I am sure it will go through. What this bill would do is require law enforcement to find out who is betting, figure out where and how they get the money from you, that you indeed bet afterwards, then prosecute you hoping and praying that entities located completely in foreign countries are going to cooperate. Sure it COULD happen, but who thinks it will? My advice is if you are on the FBI Top 10 most wanted or say a drug dealer or terrorist...maybe refrain from betting. If you are Joe Average with little else to hide, don't lose any sleep over this.
Can you just imagine the backlash if word came out that the Justice Department instead of trying to prevent real crimes that cause real injury was spending its resources trying to collect reams of data on who was betting offshore. Yeah Kyl might like that, but most Americans would tell them to get their priorities straight.