The 'throw' will have to be vertical. Without a pre-throw windup, and I have no idea how to simulate that without using an actual robotic arm, you can't do a sidearm throw.
So there's an idea - instead of making a suped up treb - why not make a cheapo robotic arm mounted on an adjustable paltform ?
With the motion capture info, you could accurately simulate throws properly.
https://www.thingiverse.com/search?q=robotic+arm&sa=&dwh=945ad72c437276ePlenty of designs out there.
The thing is that the vast majority of sling styles rely tottally on the windup to generate power and the only way to replicate that is to make an actual arm, mounted on a powered platform to simulate the waist.
Motors would be the main cost. But you can get some pretty beefy stepper motors. gears are cheap and fairly easy to make (from my point of view). I guess you could use standard arduino boards and gcode to run the throw.
Just makes more sense.
At the end of the day, we need soemthing that can move pretty fast, but on the positive side we're not using anything heavy at the end of the arm.
So let's make an actual simulated slinger, rather than something static like a trebuchet that can only replicate the release part of the throw
You need 4 joints: waist (rotating platform) shoulder, elbow and wrist.
You can use 2020 aluminium strut for 'bones'. Should be a way to make sliding joints to vary effective 'bone' length.
The mechanics are fairly straightforward. It's the software that's the bugger.