Why would that save them from the claws of FBI? Could you elaborate Lotus? Thanks. 
Personally, I think that continuing to accept US visitors and continuing to do business with US companies for domain names, hosting or CDN is silly.
As Albert Einstein said:
The definition of insanity is doing the same thing over and over again and expecting a different result.So if Bedpage hopes to experience a different fate than Backpage, maybe they are insane.

But what could lead Bedspage managers to believe that using CloudFlare could protect them from the FBI? What can lead them to believe that they do things differently now?
I do not really know. In my opinion, accepting ads from American sex workers, accepting American visitors, doing business for hosting or CDN with an American company, is continuing to do the same thing. But here are three hypotheses:
1) CloudFlare (a US company) has subsidiaries in many countries with headquarters in various cities including London. Perhaps they are convinced that their non-US subsidiaries are not subject to US laws?
2) CloudFlare is (among other things) a content delivery network (CDN). That is to say that it hosts the data of a website on more than a hundred servers (151, I believe) located on several continents: America, Europe, Asia, Africa or Australia .
So I guess they can host the content of a specific website (Bedpage, for example) on non-US servers and claim that they are not subject to US laws?
3) CloudFlare is accused of protecting unsavory organizations. Maybe the Bedpage managers think that CloudFlare will use all its stuff to protect them?
But as I said, I do not really know. What I do know is that some people have dubbed CloudFlare... CrimeFlare.

You may want to take a look at the following website:
crimeflare.org:82