It sounds like a beautiful tree, lisa--I would say don't drill it.
The risk is the same one we try to minimize by using a
tree-strap made of nylon to put around a tree, rather than looping a chain around the trunk when trying to brace a 4wd vehicle when winching, etc.
Drilling a couple of holes probably wouldn't kill it, but would create an opportunity for bacterial "infection."
Maybe get a nice, decorative wrought iron plant hangar/bird feeder hanger that you can put into the ground. Advantage being that you can relocate it if the sound is too faint or too overwhelming due to its proximity to where you are listening to the chimes while you wash your Spyder. (See, it was a Spyder question.)
My $0.02