The arbitrator telling you that you had a right to seek fees from your husband was a courtesy to you. He was telling you additional rights that you have. As for counter-suing for a refund of fees already earned by your attorney, where do you think you are going with that? If the fees have been earned, you are not entitled to a refund. The bottom line is you owe fees to your former attorney and you are looking for any way out of paying. Whether the process is fair or not, you still owe fees and the delay tactics are not getting you anywhere. Your former attorney had no choice but to sue in an effort to collect from you.
I hate to be the bearer of even more bad news, but I'm afraid it's probably too late to do anything about the fee arbitration award.
A fee arbitration is either binding or non-binding. If the arbitration was non-binding, then you had 30 days after the decision to file a request for a trial de novo. It sounds like this 30-day window has long since closed.
If the arbitration was binding, your only remedy was to vacate the award on the basis of arbitrator bias or some other ground specified in the Code of Civil Procedure. The petition to vacate, however, needed to be filed within 100 days of service of the award. Again, it sounds like that option is no longer available to you.
You misinterpreted both the purpose and the meaning of the arbitrator's advice to you that you had additional rights and opportunities to collect fees in the course of your family law proceeding. And I can't help but question whether your misunderstanding on that point colored your view of the proceedings and the outcome. If you look at the statistics on bar-sponsored fee arbitration programs (something I would encourage you to do), both overall in the state and area by area, you will find that the statistics as to client outcomes clearly undermine your inferences that cronyism is systemically at work at the expense of clients.
But to answer your specific question: yes, of course, you can plead and file a counter-claim against the complaint filed by your former attorney. But whether you "can" sue is a very different question than whether you will win. It is not unlikely that the issues will be viewed by the court similarly to way that they were viewed by the fee arbitrator. In most cases not premised on odd or extraordinary facts, that would be the expected result. And you may find it problematic to obtain skilled and experienced counsel to represent you in that counter-claim, for the very same reason.
My responses to questions on Avvo are never intended as legal advice and must not be relied upon as legal advice. I give legal advice only in the course of an attorney-client relationship. Exchange of information through Avvo's Questions forum does not establish an attorney-client relationship with me. That relationship is established only by individual consultation and execution of a written agreement for legal services.
I have been a fee arbitrator for almost twenty years. I work in a very large county compared to most bar associations, and yet I still handle arbitrations with attorneys who I know by reputation or by having served with on other bar committees or attorney events. Even in Orange County (third largest bar association in the state) it is a small world. However that does not mean that every time I act as arbitrator with someone who I have met in the past, or who has similar experiences with judges, that I rule in their favor or give them any kind of advantage.
Keep in mind fee arbitrators do not get paid. It is entirely voluntary. Do you think that arbitrators volunteer dozens of hours a year so that they might be able to help their buddies? Not a chance. We volunteer because we believe in the process and want clients and attorneys to have a less formal, less expensive and less stressful way to address the issues that come up related to attorney fees. We know that clients often feel overwhelmed by the disparity in power when attorney and client have a dispute, because attorneys have the power of knowledge and experienced advocacy. Therefore, fee arbitrators go into the process understanding that while they must be fair to both parties, they need to listen more closely and think more intently on the client evidence and arguments to make sure they have a fair hearing. In other words, it is the client that usually gets the benefit of the doubt in those hearings simply because we know that the client is not trained in legal process or the law.
I will now get off of my soapbox and address some specifics of your post.
The fee arbitration process has a very limited jurisdiction, meaning that only certain issues can be raised in it. All of the other issues that you might have had with your attorney are still available to you as long as you have not missed the time to bring them.
The bright line between what is possibly covered in a fee arbitration and what cannot be determined in that process is this: you cannot ask a fee arbitrator to determine issues of malpractice, only issues related to the entitlement to fees. Stated perhaps more clearly, a fee arbitrator cannot award you any money as damages for attorney conduct that caused you injury. The arbitrator can only determine if you owe the fees or you don't owe the fees. It is possible that you owe the fees, but that your attorney caused you damages by acts of malpractice that far exceed the value of the fees. A fee arbitration only addresses the first part of that equation. To adjudicate malpractice (i.e., attorney conduct that fell below the standard of care for attorneys causing damages to the client) you must file a lawsuit separate from the fee arbitration.
You have received some good responses that talk about the following, but I will try to state it perhaps a little differently to help you understand. When a fee arbitration is carried out, the parties are given the opportunity to elect whether the final decision would be binding or non-binding. If you and the attorney elected "binding" arbitration, the prevailing party can take the arbitrator's award to the superior court and convert that award to a final judgment. It is at that stage the non-prevailing party can dispute the arbitrator's award, but the grounds for disputing that award are very limited. However, one of those grounds is bias by the arbitrator. If you believe there was provable bias that influenced the decision, your time to raise that issue is at the time the prevailing party files in superior court to convert the arbitration award to a judgment.
If you elected non-binding, the process of disputing the award is far easier, but there are critical time limits involved. If you get a non-binding award from the arbitrator, you simply file a document asking for a "trial de novo" and you get it. You start over with a superior court judge instead of an arbitrator.
END OF REMARKS IN COMMENTS
This answer should not be construed to create any attorney-client relationship. Such a relationship can be formed only through the mutual execution of an attorney-client agreement. The answer given is based on the extremely limited facts provided and the proper course of action might change significantly with the introduction of other facts. All who read this answer should not rely on the answer to govern their conduct. Please seek the advice of competent counsel after disclosing all facts to that attorney. This answer is intended for California residents only. The answering party is only licensed to practice in the State of California.