loud noise on starting enging (and on restart)

I have a 2006 Prius, package 5. I get a loud noise--almost like a loose engine mount or loose connection in the exhaust system immediately after starting--and for a while each time that the gas engine shuts off (to go to the electric motor only mode). The sound seems to be in the engine compartment. I open the hood and I'm clueless.
This will persist for the first 2-5 or so minutes of driving (sometimes less, sometimes more).
It isn't there on the expressway at 55 mph--and but it is noticeable if I'm parked at a drive-thru window or sitting waiting to pick some-one up at a drive thru.
It isn't heard once fully warm with stop and go for construction work on the express-way.

I don't have a single dealer in the area who I can trust for work. (Buffalo).
I have tried to get mechanics who I have used previously for non-hybrids to look and no one wants to touch the car.

I am a basic klutz and probably shouldn't be owning a three year older hybrid. I thought about trading it in but bought the extended warranty instead.

Should I change the 12V batter just to be safe?
I am afraid of this car breaking down after reading posts on inverter coolant/batteries/transaxle.

I did buy the 7/100000 warranty but it seems, based on other posts, that it isn't worth much.

 

It's there more when cold. I can't always have it happen but it happens more often than not when the engine is cold.