I think you need to shift to another API option : HTTP API, so that you can have a direct private integration with your internal ALB. This is perfect solution for you.
The below AWS post answers all your query
Thanks & Regards,
REST APIs only support VPC Link to a Network Load Balancer (NLB). ALB's are not supported. You can either use HTTP API which do support ALB (in case the current feature set of HTTP API is appropriate for what you need) or you can use a solution Like this: API Gateway -> VPC Link -> NLB -> ALB -> Application.
You can find more info here.
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