In short: not easily.
Long version: For small trebuchets, they are generally durable enough that if you strike the front of the track on the follow through or on a miss fire, it won't break anything, so you really only need to consider how much track it needs to throw properly. (For GBB, the track needed to deal with a miss fire is perhaps 4-5 times longer than used on a normal throw, which is the main reason I think FAKAs scale up to large machines poorly) How long that is depends on your secondary trigger timing, which (when properly tuned) depends a lot on mass ratios, friction etc. You are aiming for a 100:1 mass ratio (great choice, though FAKAs actually can work with higher ones pretty well unlike many designs), so I'd guess your tuning should be similar to my GBB at 10 pounds CW, which used less half the length of the upper arm (called the Short Arm (SA)) + hanger of motion on the track. If you don't have a proper secondary trigger, or you opt for tuning like j-buchet (which lacks an upper arm stop), you will need to allow for motion around or a bit more than your hanger+SA. With an upper arm stop, and a well tuned secondary trigger, you can get away with a track thats much shorter, around half the length, or maybe even 1/3.
Note: these lengths are the amount of displacement needed, so add the 1" carriage lengths to that. Getting the tuning correct is easier if you don't tend to hit the front, and depending on your specific machine, you might need a bit of extra length, so I'd go with an 8" total track minimum, but make sure your machine's base is bigger than that. GBB tended to rock on its square frame. If you don't have an upper arm stop+secondary trigger (not recommended, but people seem to do this anyway, and it does work) you will want something more than 9 inches (example applechunker)
Since you have made a FAT, you probably understand the carriages reasonably well, so this applies to you less than others, but I still recommend a KA on wheels like mini-ka. It is almost the same on a small scale since frames can be really light for small machines, except you don't need to worry about track length, and its quite a bit easier to make. That said, I would love to see a well made small FAKA. If it works, please make a machines page for it with some pictures and ideally video. It would make a great addition (so would your similar FAT).