Hang tight here folks, I feel like I MIGHT have found it.
I just found this thread from a Kawasaki singles forum.
Now, look at my carb. It is the same hole as this photo, yes? The thread states it as a very common fault, and for a bike that "will not run when hot" and "air gets into the cylinder". This could be the leak I've been looking for?
*I'm sorry I bit my nails. It's the Kawasaki's fault.