This will probably get the anti-lawyer folks up in arms, but here it goes:

A lawyer's primary duty is to his or her client, not the public and certainly not a potential defendant.  Except for certain limited instances, a lawyer is not required to fact-check a client to ensure that they are on the up and up.

The exceptions to that are:  (1) when a lawsuit is filed, because courts do not want to waste their time dealing with half baked  unsupportable claims, a lawyer must conduct a reasonable investigation to confirm facts and law, (2) a lawyer cannot provide advise or assistance to knowlingly aid a client in fraud or the commission of a crime (with apologies to Saul), (3) a lawyer must be honest and civil in deals with third parties (a lawyer could not lie in a C&D; for example, could not say "I have reviewed the Amazon agreement" etc.)

There are a few other very limited exceptions but those are the big ones.  That is **only** the ethical duty, however, not whether its a good idea to proceed to a C&D on the word of your client alone.

As far as whether it was a good idea for BM to proceed to a C&D without conducting an investigation, that is a different question as I said above.  It is **not** a good idea for three reasons at least.  

First, it does a disservice to the client IMHO.  When the bluff is called, the client (and law firm) lose a lot of credibility when they fail to follow through with a lawsuit.  Clients frequently ask me to send C&Ds to try to bully smaller companies, and I always say the same thing: I will only send a C&D if you are able to provide a sufficient retainer to pay my fees throughout a lawsuit, and you are ready, willing and able to initiate a lawsuit if there is no compliance by the deadline.  No bluffs, they are bad for the client's and the firm's reputation.

Second, sending a C&D without an investigation for any kind of internet activity risks a major backlash (as happened here).  It is a pretty common defense to simply post the C&D for the world to see, and let the public response do its job.  Absent a real wrong, most people hate it when lawyers send C&Ds (especially to journalists or web sites) .. doing so almost ALWAYS backfires badly, and I personally have dissuaded several clients from making this mistake.

Third, for the reasons above, if I send a C&D client I need to be ready to follow through with the lawsuit.  Which means, I won't send a C&D without first making sure I can sign my name to a lawsuit.. which in turn means I have to conduct an investigation anyway because filing a lawsuit requires such.  But that is just my practice (and the practice of most top lawyers I know), not an ethical requirement.