Well you really can do anything you want with your Keystones...jazz, swing, metal, rock, funk, pop, jug...you won't be hindered from playing jazz just because they are more "rock" sizes. I'm sure it would make manufacturers happier if everyone had to have a separate kit for everything. But what is a jazz drum kit anyways? Catalogs today will have you believe it's a 16-20" bass drum and smaller fusion toms, but look at what they played back in the day. Monster bass drums and normal toms. If you think your shells are too thick then that could be discouraging, but I've seen lots of jazz played on Sonors.
Tune your toms and kick higher than you would for rock, and maybe tune your snare to a more medium tension. Whenever I break down to a four-piece to (attempt to) play jazzier stuff, I tune my toms up until they "sing" and the kick drum up until it's a little bumpier. My snare is usually pretty cranked so I drop it down to get a fatter sound. I've heard tell that some jazzers like coated batters on their kick, like a coated emperor. A coated head on the front might be a good idea, too. Don't pillow it up too hard if at all; try the old felt strip thing. Some people like to use fiberskyns all over the kit. I have one on the front of my kick and I dig it.
All of that is just a vague nudge in the direction of what the average "jazz" sound is. There are guys out there playing jazz on 12 piece kits with clear G2s, and they're probably making good music. Experiment with different tunings a lot and try to find what YOUR "jazz" sound is.