Sadly, airlines get their visa information from a database run by a company that predictably does not know very much about Taiwanese immigration regulations. It’s the same database Cathay Pacific use, and they also insist on return or onward tickets. I argued with them and eventually they showed my the law. It stipulates passengers must have a return or onward ticket, but it does NOT say that the passenger must show a confirmed reservation on a flight or ship within the validity of their visa. An Open ticket will do. Therefore AA are out of order. Point this out to AA and get them to guarantee you a reservation on the date you originally had reserved. There is some chance that when you get here there will be no seats at your price for your date. Get AA to put in writing that they will give you a seat even if your bokoing class is unavailable as I assure you that they will try and get another few hundred US out of you when you come to change it.
Alternatively, buy a one way FULL FARE fully-refundable Cathay Pacific TPE-HKG ticket either Open-dated, or with a reservation for some date acceptable to AA. When you get to Taiwan, refund the ticket. I recommend you buy it from Cathay Pacific in Taipei (who will be doing the refund). If you use a credit card (as you will have to) any refund will take a month or so as it will be done in HK probably, but whatever way it is done it will be in full, minus NT$700. If Cathay Taipei won’t sell you a ticket over the phone, try Cathay New York (or wherever) but be aware they might have to send your ticket back to where it was issued. They might issue the ticket in TPE for you, and send you a confirmed itinerary which should be enough for AA to tick the boxes on their paperwork. A pain in the arse, but as long as you get a full-published-fare ticket you will get the refund eventually wherever the ticket is issued. The last full-fare o/w TPE-HKG I bought was NT$7,800 btw.
What a hassle. Almost easier to come on a landing visa and doing a visa reun to HKG when you have a bullet-proof reason for being in Taiwan like you do.