Yeah, after thinking about it a little differently, I plugged my personal Tele (with a traditional tele pickup) into my amp, dimed it, and started tapping the bridge with a screw driver. The tapping is loud, like one would expect, but it's not an annoying sound, and there is no ring, it's a solid sound. Then I plugged the problem guitar in, and did the same thing. It's equally loud, but there is a ring underneath every tap, I guess you would actually describe it as a harmonic or overtone.
So, paying closer attention to the bridge and this particular guitar. The corners of the ST59 are touching the bridge. The 4 anchor screws are on the verge of being stripped. The front edge of the bridge is slightly bent, and you can hear it click against the body of the guitar when you tap it.
So, I realize this all sounds opposite from my previous posted experience, but I think I was looking at everything trying to make the pickup the culprit. I couldn't actually mount the other bridge I tried as the screw pattern didn't match the guitar. And I heard the tapping and assumed it was the same, but I wasn't listening for that ringing sound specifically.
Perhaps this is a perfect storm, a slightly noisy pickup combined with an ill fitting bridge?