I have used synthetics for years in all my cars - engine oil, trans fluid and gear oil. I changed the oil in the 400 convertible I just bought to synthetic the first weekend I had it.
I have used AMSOIL (was a dealer for years - not sure if I still am.) in the engine, trans and differentials.
I have also used Mobil 1 - you can't go wrong with either one.
Thanks again for the hood. Hope you were able to cool off those Bud Lights when you got home.