Observations:
1. Someone hacked off the usual Molex connector and hooked it to some 3-pin .1" Dupont connectors. (an odd pin arrangement

)
2. The green wire with the ring-crimp connector connects to frame ground. It is there to bleed off the static electricity charge generated while rolling the trackball.
3. There is no optical encoder in your pics (Opti-Wiz, Opti-Pac, Mini-Pac, I-Pac 2015, Ultimate I/O, etc.) so I'm not sure what it was hooked to in the old cab.

To get your Happ trackball working:
The easiest and least expensive way is to use an
Opti-Wiz (no-solder) and the adapter cable
here that connects from the optical "red boards" to the Opti-Wiz. (replaces the current hacked harness

)
The bad news:
1. The U-Trak and Happ trackball mount in
completely different ways.
2. Since you have a plexi cover and since the artwork is already stuck to the wood, you won't be able to hide the Happ mounting plate.
The bottom line is that you need to decide between two options:
1. Buy a U-Trak/optical encoder(unless you already have one of the ones listed above)/RGB LED and use the Happ for another project/sell it. (more expensive but easy to install)
2. Buy the Opti-Wiz + adapter cable and modify the CP so you can use the Happ. (less expensive but much harder to install and the mounting plate will be visible)
Scott