I suspect they didn't want a bladder, as constant pressure is actually bad here. Using a spring means that you can get more force the harder you pump.
I'd also say it's more difficult to use a bladder for such small volumes. I'd be interested in making a similar gun that used a bladder instead, but you can't really control the duration of a burst with a gun like this. I'd prefer having something that doesn't use water unless you want it to.
I'm thinking they used a spring because an air pressure design must have been patented. As I've said, I've seen one - it's actually in my backyard because my neighbor left it out (he doesn't really care for the gun). It's performance is terrible, but I'm sure BBT pulled it off properly.
