Thank you for responding and clarifying your statement. The question is not really how do you write the complaint letter, it's what you are complaining about and what are your remedies.
You are stating that an attorney represented RBC in an arbitration proceeding in California wihtout being authorized to practice law in the California, per the State Bar. This arbitration proceeding had a negative outcome for you. Before we move on from the arbitration, you should understand that arbitrations are sometimes appellable and the arbitration decision is unenforceable. I'm unsure from your previous emails whether or not you have attempted contesting the enforcement of the arbitration provision or the arbitration award, as you indicated that a California court has ruled they are not the correct forum (this implies that they compelled you to arbitration, but it also could imply that the court things a different court in another state has superior jurisdiction over the issue). I may be looking at something that you have already had analyzed, but I would be remiss not to mention it just in case you have not.
That being said, the main complaint you've brought up here is that the attorney was not authorized by the bar to act in the arbitration. You want to write a letter demanding a "settlement" in exchange for not making a complaint about the unauthorized practice. There are two problems with this. The first is that you are proposing sending the complaint to entities which have no oversight over this sort issue and where your complaint would land on deaf ears. Writing the New York Stock Exchange about one of their member's use of an attorney who was not licensed in a specific forum is not going to trigger any action. Writing Congress may get you a response that says "take it up with the Court's or with the State Bar". Writing the SEC will likely be ignored (I brought a fully researched case to the SEC regarding the defrauding of US shareholders in the amoutn of $65 Million which was simply put into a large stack and I was told they "did not have the time or resources" to handle any further complaints or prosecutions). Further, the SEC only has jurisdiction to enforce securities laws, which the unauthorized practice of law does not violate.
The second problem is that a the unauthorized practice of law is a criminal action. You've mentioned previously that you understand that you can't threaten criminal prosecution in demand for a settlement. This would be criminal extortion on your part. Yet, threatening to report a person for the unauthorized practice of law to any governmental agency which could "punish" the lawyer or the company would also expose you to allegations of extortion and blackmail.
There are four remedies for the unauthorized practice of law: (1) an injunction; (2) criminal prosecution; (3) criminal contempt; and (4) quo warranto writs. An injunction would have been useful to you if you would have asked for it during the arbitration. You could have moved to a California court for an injunction against the RBC Minnesota attorney from practicing law in California. This may have stalled the arbitration until RBC hired a California attorney to take over. Criminal prosecution and criminal contempt are both things that a DA or the State Bar can use to enforce, but not a private party. Writs quo warranto are enforcement proceedings brought by the State.
Further, the breach of duty by the lawyer was not of a duty owed to you, it was his duty to RBC which he breached. If anything, RBC could sue him for violating the attorney-client relationship by taking such actions.
So let's assume that you send a letter to the attorney stating that you are going to file a complaint against him for the unauthorized practice of law with the California Bar and the Minnesota Bar, the only entities that have any enforcement power. Let's further assume that you state you will not make these complaints unless some sort of compensation is given. The attorney may look at it and say "I'll throw the guy some money to make him go away", or he could call the police to report an attempt to extort him. My conclusion on this matter is that you should simply report them, instead of demanding money to not report them.
Further, I think you need to readdress your appellate situation with FINRA. You CAN refuse to abide by an arbitration ruling on the grounds that the arbitration was illegal. The other side then has to move to enforce the arbitration in the courts. At that point, you would bring up legal arguments regaridng why the arbitration was invalid (i.e., it went forward without authorized attorneys and they did not pay the required arbitration fees, OR the underlying arbitration provision was unconscionable and not supported by consideration and thus unenforceable; OR the arbitrators did not apply the correct law so that there was no due process). These are all possible avenues of protection for you under the Federal Arbitration Act. Again, I'm not certain that you have had your arbitration lawyer look over these possibilities, but I would not be doing my job if I did not bring it up as a suggestion.