I have two RT97S repeaters that I've (unsuccessfully) attempted to attach a controller to. My experience is the exact same as yours with white noise being transmitted by the repeater whenever the control board keys the repeater. I was trying the ICS Basic and Linker IIA controllers.
I tried:
Putting a reed relay to key the repeater
Running the repeater and the controller off of different power supplies
Isolating the repeater's COR from the controller with a transistor switch
Isolating the repeater's COR from the controller with a diode
Isolating the repeater's COR from the controller with an optoisolator
Disconnecting the receive audio from the controller
Various combinations of everything above and probably some other things I'm not recalling at the moment
I had to move on to some other projects, but plan to getting back to trying. I'd bought a pair of the ID-O-Matics, thinking that maybe the issue was with the ICS' transmit audio interface circuit (both models of controllers have similar circuits). From your post, it's probably not the issue though.
One possible solution that I didn't try (yet) was coupling the audio through transformers. Combined with optoisolators/reed relays, that might totally isolate the repeater from the controller.
It's possible that the white noise that is transmitted when connected to the controller is actually the unsquelched receive audio. I've tried changing the squelch level as well as running ctcss, but it makes no difference. If it is the receive audio, that may be a show stopper since there don't appear to be any schematics available.
I didn't have access to my service monitor when I was trying to interface the controllers, but an interesting test would be to generate a signal with a 1KHz tone on the repeater input and see if you hear the tone instead of white noise.
