Does the sound ever go away, like if you drive long enough for the entire exhaust system to warm up fully to normal operating temperature?

If the noise does eventually stop, perhaps it's coming from where two pieces of the exhaust system meet... and there is a very slight difference in expansion rates between two pieces of metal at a pipe connection... it might take a long time before both pieces reach their full temp, and so ending the tink.

That begs the question: does the tink sound also happen when the engine is off and the exhaust system is cooling down? If my conjecture is correct, it won't necessarily tink in reverse, but it could.

So maybe a recently opened pipe connection isn't fully closed?