Thanks so much for the help and suggestions guys! I ended up going with the two chamber reactor design that Omicron came up with (I'm so glad you were able to figure it out), and here's how it looks:
It's perhaps not the most space efficient automation system, I could have used AE to do it, but I wanted to branch out, and I really like what I came up with. The reactor is controlled by another Machine Box with a Redstone Transmitter in our main control room (the reactor room is hidden far below ground. We were afraid of explosions). We currently go through about 1 uranium every 5 minutes, so our quarry can't entirely keep up with just these four reactors, but it's close. If we had increased uranium input, then I could double the size on the floor, and then it would be just two more layers above to double it again (they could share the buffers that are displayed on top in the screenshot). I don't think we are anywhere close to producing 4 uranium / 5 minutes though, so that might be a while (especially since we can't craft uranium from UU matter, not that it would even be cost efficient).
Now though, I have 496 Thorium and 66 Plutonium sitting in the ME network waiting to be used. I think I might try to get a computer cube to try and make some designs from those, or just run with the suggestions from Omicron (I have no clue how you can reliably come up with GT designs without the Java planner, you have some kind of magical mojo).