Just wanted to mention that my battery cover arrived today and it fits my GRE PSR-300 perfectly. It is a slightly darker shade of black than the rest of the scanner, which is no biggie.
To recap, the battery cover and the battery holders for the Whistler WS1010 (
https://whistlergroup.com/collections/scanner-accessories) fit the GRE PSR-300 scanner as well as the RS PRO-164 scanner. This is no surprise in retrospect because Whistler bought the assets of GRE (who also made scanners for Radio Shack).
It's possible they also fit the PRO-97 but this hasn't been verified.
Also, just to note, although it probably won't be of much interest to people on this forum, the non-swivel WS 1010 belt clip also fits the Whistler TRX-1. I got it because I found that the swivel version that comes with the TRX-1 was too bulky.