How to get private Elastic-Beanstalk Environment DNS Endpoint?

My setup is the following: i have a VPC consisting of 3 subnets: public, private with NAT and isolated. I have an elastic beanstalk (single instance) environment running in the private subnet with Nat and it works.

What i want to achieve: have a private hosted zone with a record pointing to my private EB env.

What i'm struggling with: the environment DNS of my EB env points to its public ip (which is not reachable since its in a private subnet and thats good). But to create a DNS record pointing to my private EB env i need a private endpoint of my env, right? How do i get a URL like the public one for my env (e.g. myapp.eu-central-1.elasticbeanstalk.com) but resolving to my private ip, not the public one?

The answer of Matt was very useful to me, it worked. But I had to make some changes:

commands:
  01_update_r53:
    command: |
      IMDSTOKEN=$(curl --silent -X PUT "http://169.254.169.254/latest/api/token" -H "X-aws-ec2-metadata-token-ttl-seconds: 21600")
      IP=$(curl --silent -H "X-aws-ec2-metadata-token: $IMDSTOKEN" http://169.254.169.254/latest/meta-data/local-ipv4)
      AWS_R53_APP_DOMAIN=$(/opt/elasticbeanstalk/bin/get-config environment -k AWS_R53_APP_DOMAIN)
      AWS_R53_HOSTED_ZONE=$(/opt/elasticbeanstalk/bin/get-config environment -k AWS_R53_HOSTED_ZONE)
      echo "{\"Comment\":\"Update IP\",\"Changes\":[{\"Action\": \"UPSERT\",\"ResourceRecordSet\":{\"Name\":\""$AWS_R53_APP_DOMAIN".\",\"Type\":\"A\",\"TTL\":60,\"ResourceRecords\":[{\"Value\":\""$IP"\"}]}}]}" > awsr53.json
      aws route53 change-resource-record-sets --hosted-zone-id "$AWS_R53_HOSTED_ZONE" --change-batch file://awsr53.json

This needs to assign to the service role used by the instance permissions to perform an upsert on Route53. I suggests to perform the script mentioned before via SSH first to debug