First of all if the arbitration clause allows for reconsideration before the arbitrator then it is definitely allowed. Also, if both parties agree then they might ask the arbitrator to reconsider or clarify the award/ruling. If there is no agreement on this then you will find the next paragraph relevant.
I. Fees and Costs Issue
In the absence of mutual agreement the party seeking fees and costs would first go back to the arbitrator and argue that the award was not complete. The main obstacle to this is the common law doctrine of functus officio (office performed) meaning that the arbitrator's jurisdiction over the case ended when the final award was issued. However, final is the key word here. One California case, Jannis v. Ellis, 308 P.2d 750 (Cal. Dist. Ct. App. 1957), adopted the view that functus officio only applied to awards that were final, complete, and coextensive with the mandate submitted to the arbitrator. (the 9th Circuit also holds this view) This is the dominant viewpoint nationally and allows arbitrators to correct a mistake that is apparent on the fact of the award, finish an incomplete award, and clarify ambiguity.
In this case it appears that the award could be incomplete without an award by arbitrator on costs and fees. This would allow the arbitrator to revise the award to include fees and costs.
II. Judicial Review of Arbitration Awards
Under California law any party to the arbitration, after proper service/notice to the other parties, has three months, unless the parties extend the time in writing, to seek a confirmation, correction, or vacation order of the arbitration award from the Superior Court located where the arbitration took place. The court must grant this order of confirmation unless the court finds that the arbitration award was the 1) result of corruption, fraud, or undue means, 2) the rights of any party were substantially prejudiced by the misconduct of the arbitrator, 3) the arbitrators exceeded their powers, 4) a party was substantially prejudiced when the arbitrator refused to postpone a hearing after good cause was shown to do so or 5) where the arbitrator refused to hear evidence material to the case, or 6) where the arbitrator should have been disqualified from hearing the case on grounds elsewhere in the statute. In those circumstances the award must be vacated. There are also circumstances where the award could be corrected by the court where there is an evident miscalculation or mistake, the arbitrators exceeded their powers, or the award is imperfect in form without affecting its substance. The court can send vacated awards back to the arbitrator or a new arbitrator for further proceedings or correction. Confirmation by the court gives the award the same status as a civil judgment by a state court. If the award is not confirmed (i.e. none of the parties seeks confirmation) then the award is still binding under contract law unless it is vacated.
In this case you would have to be more specific about what is meant by non-disclosure of the arbitrator. If it was just a case of not disclosing the identity of the arbitrator to the other party then that does not seem to fit the criteria for vacating the award. If there was non-disclosure of some fact about the arbitrator that might indicate that the arbitrator was biased or some other problem that substantially prejudiced a party then it may well be vacated if a party were to ask the court to do so and follows the procedural notice requirements.
As far as attorney fees and costs of judicial review in Superior Court those would probably, depending on the circumstances, fall under the contract's prevailing party clause just like the other fees and costs in the arbitration, but the parties could obviously agree to an alternative way of dealing with the issue.
Other circumstances could also be relevant to these issues and the legal information contained herein.