This looks like a standard engine failure on take off.
Engine surges due to a breakdown in airflow through the compressor.
The crew do the right thing. Fly the airplane until at a safe altitude and then deal with the engine problem.
After the engine is shut down fuel is dumped (in an approved dumping area) to reduce the weight and then return to land.
This is what we, as aircrew, are trained to do.
I don't see why the FAA or NTSB would be involved in the investigation.
Here in Europe the crew would file an Air Safety Report, and later maintenance would do the same when the cause was identified. These are sent to the regulatory authorities for their information and consideration as to weather follow up action is required.
Safe flying.