Well I don't know specifically about flaming river, but I can tell what I think and other can chime in.
If you short point 1 to ground, it should toot the horn.
This is the cancelling cam. Point 2 touches point 1 and remains in contact when you turn the steering wheel.
The bottom of this tower on the cancelling cam is point 3, and it is the other side of the brass ring shown in the previous picture. You insert the nail and spring device into the tower. So the end of that wire, if shorted to ground should make the horn toot.
OK with this steering wheel you don't need the wire attached to the nail and spring. The tower of the cancelling cam should stick up through the steering wheel, there is a hole in the steering wheel to accept insertion of the cancelling cam tower. On the underside of this convex spring disc is point 5 where the nail and spring device makes contact. Now finally, when the convex spring disc is pushed down, it will make contact at point 6. When the convex spring disc makes contact with the hub right there next to the steering wheel mounting nut, it shorts the electric connection to ground, and that causes the horn to toot.
So here's where it is not clear to me. I don't know if you need a stock cancelling cam with its nail and spring guts (and there would be no wire). Or, if you need to use your flaming river cancelling cam with nail and spring guts and remove the wire or cut the wire most of the way off and just leave enough exposed and stripped wire sticking out of the top of the tower to contact the underside of that convex spring disc. Others here on the forum may know that answer. If you have a stock canceling cam, give it a try, and if you don't a posting of a want-to-buy ad in the parts section will probably fix you up with one cheap.