I have set up an Amazon EC2 Auto Scaling Group that is based on a Launch Template using its latest version. The Launch Template uses an AMI. I have also set up a Data Lifecycle Manager that makes a back-up every X hours so that my data is up to date.
How can I "feed" the AMI ID to the Launch Template dynamically so that everything is automatic?
You can't. Launch templates are immutable. You have to create new versions of your template, and then modify the Auto Scaling group to use the new version.