Send Success message from Power Automate to Power Bi report

Unfortunately that is not yet possible. You need to use a semaphore in Power Automate to prevent subsequent requests from causing harm.  (When the flow starts check if the semaphore exists. If not,  create it and run the flow. Remove the semaphore when the flow is done.  If it already exists that means another instance is running and the new instance should exit)

The Power Automate action is blind - all it does is request a new flow run. It has no idea what happens after that.  Hopefully that will change in upcoming releases.