Consider flying out of LAX, there are lots of flights (Eva, Korean Air/Delta, and Cathay/American, in order of most expensive to cheapest). We flew Southwest to LAX, picked up our luggage, and walked over to the International Terminal.
Try calling the private Urgent Care clinics around you - they typically contract with private and university labs that can get you a COVID result in 2-3 days. Taiwan relaxed their COVID test requirement to be within 3 business days of travel, so if you get tested on Thur, you have until Tuesday to depart.