Yes you can. If it's based on a written contract, you have 4 years to bring th claim. The state bar is fee dispute proram is the easiest way for you to do this. Good luck.
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I agree that you can take this attorney to fee arbitration, but you could also sue this attorney in small claims court. Sometimes that is faster and easier. If you have a dispute with the attorney and he knows it, he should provide you with a form, "Client's Right to Arbitration". That usually comes when the client owes the attorney money, but since it is the other way around according to your letter, you may go to small claims court. The limit is $7,500 to sue individuals and $5000 for corporations and other similar entities. Good luck.
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