Common gotcha. Allowed Origins is an exact-match list, not a wildcard. You need to add both:
If you're hoping for https://*.mysite.com - that's not supported on any plan right now. API Gateway can't return a wildcard Access-Control-Allow-Origin when AllowCredentials is on, which the widget session endpoints require. The long-term fix on the Jarvis side is to move the embed endpoints to a Lambda Function URL (which does support wildcard CORS), but it hasn't shipped yet. For now, enumerate each subdomain you need.