First, I do not think it has anything to do with the transmission cooler, as I have changed radiators, trans filter, and fluid with no difference. Ditto on oil filters. Also not HVAC.
It definitely is temperature related; didn’t occur during the summer, seems like 50f is when it is apparent. If really cold, no moan until engine has run for a few minutes, and longer time to occur and longer it lasts the colder it is.
Personal opinion is that it is the oil pressure regulator, as NA spec uses bypass oil cooling- when regulator opens, it directs flow through the oil cooler, which is rigidly mounted to the body and why some of you indicate that you feel it. I had some luck with a 3.3L Chrysler engine with a similar noise by grinding a groove in the regulator piston.