No. There is no 3-person arbitration. This is just the start of your case, and the Admin is asking if you are representing yourself so that they can file it with the Pre-Se Admin staff. I would simply email back and state that I am representing myself pro-se and that this is a CONSUMER arbitration matter.
He can object, and you can respond to the objection demanding an in-person hearing citing the consumer rule regarding such. Also state that you intend to ask Barclay's to present a live witness with first hand knowledge of your disputes and on-going issues with your account.
Researching the laws and knowing what laws they violated should have been done well before filing the case. Regardless, you must do this NOW. You need to start working on a full brief that clearly lays out what happened, what actions you took and what actions or inaction THEY took and how those actions or inactions violated which laws. Both CA and Federal laws. Some laws to look up would be any California Unfair Business Practices laws (sometimes referred to as UDAP laws), the Federal Fair Credit Billing Act, the Fair Credit Reporting Act, and potentially the overly complicated Regulation Z part of the CFB (Truth in Lending Act).
Since it is early, you have at least a month, so I would put together a formal brief and have it ready. The first step is that Barclay's will get a $5k bill from AAA. They will pay it. Then an arbitrator will be appointed and contact you to set a date for an initial phone conference. The initial conference call is to set dates for discovery and the hearing. During the call, ask the arbitrator for permission to submit a formal brief outlining your full dispute. You can ask for 30 days or so to have it submitted if you need more time to complete it. He SHOULD grant that. And he will also give the other side time to file a reply to your brief. After that, you need to figure out what discovery items and questions to send to Barclays to prove that the charge was not yours and that they did not refund any amount you paid in error and start to have documentation to back up your brief that can be presented at the hearing.
Because of COVID, I would expect that an in-person hearing will likely be denied, but I would still push for at least a Zoom call hearing.