Consider the possibility of a leaking coolant tank. Even the smallest of leaks on this oddball part can cause overheating.
My son and his wife owned an LS that had the same symptoms and this car even baffled me a bit at first look because, in theory anyway, a tank with a slight leak should not cause overheating unless the coolant level drops too low. In this case, not.
You will also notice the tank is buried big time inside the left front fender and it’s just about impossible to see any leaks.
This may not be the problem at all; just pointing out a real oddity on these cars.