Missing the point
It seems to be simply misdirected. If you want to censor the information then you should pass a law allowing you to censor the source of the information. Why would you censor the middleman? The only reason I can think of is that censoring newspapers has a long history and is unpopular while censoring search engines is new and you can get away with it.
You mention you can still get the information from lexis-nexis, is that not still a search engine? Would it not be forced to comply with the same requests?
I understand that the crux of this debate is the law already exists so it must be followed now (which it is) but why is it a problem to try and change those laws. That is the point of democracy no? A process to change broken laws?