Congress just has to change the laws to make the SS numbers accessable via a checking devise (they can do that without giving out any information other than "yes" or "no" just by keying in the name, number and birthdate and then get an answer back "yes" it matches or "no" it does not.
Wouldn't be foolproof but would eliminate 95% of the fraud since you could probably manage to match two if the answers but would be hard pressed to get all three. By giving a simple "yes" or "no" answer, the government would still be protecting the basic privacy of an individual because it would not be giving out information, merely confirming information the individual willingly gave out.
Your probably right that Congress would have to sign off on this but thats a technicality and only one vote away. We need to put pressure on our reps to get it done.