Running an update to add a path to a healthCheck I hit the following.

Diagnostics:
  aws:lb:TargetGroup (api-tg1):
    error: deleting urn:pulumi:api-server::api-server::aws:lb/targetGroup:TargetGroup::api-tg1: Error deleting Target Group: ResourceInUse: Target group 'arn:aws:elasticloadbalancing:us-west-2:353450002364:targetgroup/api-tg1-2bf40c6/0382bcb6f8b3b67e' is currently in use by a listener or a rule

Is there a right way to pause a listener or rule so its not in use?

No, no way to pause a listener/rule. You might need to use https://www.pulumi.com/docs/intro/concepts/programming-model/#dependson to influence the dependency graph.