The idea is to get the tag to light up when it gets within range of the reader.
Nope. The RFID tag generates a very very tiny current, not nearly enough to light an led.
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agreed. if you used maybe a <1mA led you MIGHT get something, but it'd be barely visible, and definitely impair the chip's function.
I think your best best is something like https://www.instructables.com/id/RFID-Reader-Detector-and-Tilt-Sensitive-RFID-Tag/
I downloaded that PDF to see if I could do something with it but its much larger than what i'm looking for.