With API Gateway the resource has a formulaic domain name, but in Route53 that is hidden because you're creating an "alias" record. I'm not sure the same thing would apply to Lambda URLs.
It's not exactly what I'm talking about, which is unfortunately private source, but here's a close public example of something similar (MX records rather than A records). This template get's instantiated in each supported region (a small set, because SES email receiving is only supported in a handful of places):
https://github.com/mlhpdx/email-delivery/blob/master/templat...