You're really getting to know that machine Travis! I can't help with how that cable is connected, I'd guess it's some kind of clip, like a motorcycle. And if you can find a manual from that era, most manufacturers tend to use the same method from one model to the next. But depending on the routing of that cable, and how much it winds between components. I'd think about tying, and then taping some sturdy cord to the end, or ends. That way you can feed the new one back through the same way it came out. Sometimes that stuff can be a 3 dimensional puzzle. The cordage can "sometimes" make it easier. Not always, but there's no harm done, if it doesn't work.
FWIW my buggy has a short cable, that goes to a solid yoke type linkage. So it pulls two rods, one going to either side, where it looks like it then becomes cable again, and goes to each drum. That doesn't sound like what you've described, so a picture of mine from the manual would likely be useless. At any rate, I looked but didn't see any detail on the type of fastener. It just said " remove cable from grommet". There was no useful picture that I could see. All I can say, is it seemed to indicate the back, outside of the plate. And if memory serves, that's how it was on an auto, with a clip holding it.
A couple cans of walmart brake cleaner is good to keep on hand. I use that stuff a lot.