I have created an AWS Lambda with the new function URL feature enabled.
Since the URL isn't that easy to remember, I would like to create a Route 53 alias like lambda.mywebsite.com.
There is no Route 53 alias for Lambda function URLs in the drop-down menu for aliases in Route 53.
How can I call my AWS Lambda function URL via a custom domain?
Is a CNAME record the way to go?
How can I call my AWS Lambda function URL via a custom domain?
If you must use an AWS Lambda function URL, fronting it with a CloudFront distribution with your desired custom domain name is the only way currently.
There is no support currently for a Route 53 alias record, as the Host header must be set to the Lambda function URL domain. If the Host header does not exist, HTTP 400 Bad Request is returned & if it does exist but is not the correct value, HTTP 403 Forbidden is returned.
They are meant to be the simplest & fastest way to invoke your Lambda functions via a public endpoint without using other AWS services like API gateway, so a native lack of support for a custom domain name makes sense.