2015 Prius C 99k: engine rattle on start and acceleration first 15 min

Hi!

My 2015 Prius C has a worrisome rattle/flapping sound when starting, idling, and accelerating for the first 10-15 or so minutes. It is definitely coming from the engine and is audible inside the vehicle as well as outside.

I did some research and came up with a few potential causes:

-spark plugs (that would be great)
-EGR Valve / System clog
-failed head gasket

When I took the car into the shop upon purchase, the shop was unable to replicate the sound - probably because I had to drive a while to get there. When I took to the dealership, they informed me of a timing seal leak/seepage that would require a $2k+ repair. Although, I am not sure this is related to the rattle.

I have not noticed any coolant leakage, so I am fairly sure it is not a failed head gasket.

Could anyone shine some light on the EGR valve/system possibility?
I have great fuel economy, no smells, no smoke, no nada except for this rattle.

To describe the sound even better, it almost sounds like a flapping? I feel it is definitely related to the passage of air through some component and possibly temperature is a variable (due to the fact it fades when the car has been running for some time)

Thank y'all for your time and consideration.

 

Another Prius C driver listed this sound as a "mystery ticking" which also increased during acceleration. I do believe we are referring to the same phenomenon.