well, I was hoping that someone more knowledgeable might answer, but I guess the Arizona gardening forum people lost their computers in foreclosures or something. So this is mostly my opinion.
First, a lot of Sissoo trees tend to grow crooked. It's more genetic than having to do with it looking for light...etc. I have one by my driveway and it is growing towards the driveway instead of straight up. I tried to stake it but it still wants that way, and actually that's fine with me. That's exactly where I want more shade anyway.

However, just because it grows in more in one direction, I don't think it means that most of the roots go that way. The roots go wherever they can find water and nutrients.
I think if you put the emitters spaced evenly in a circle around the tree, the roots will find that water .
I also think that it would be a good idea to do something to help the water go where it is needed instead of letting it all flow to one side. I think the basin under the dripper to keep the water from flowing downhill is a good idea, but I hope that by saying 12" you're not talking about the depth. You shouldn't need more than a couple of inches deep). How fast does this thing drip?

greendesert is correct - the root zone will form a general circle, regardless of the canopy. Place the drippers evenly around the dripline.

If the water is actually flowing downhill and not going into the soil directly near the dripper, then yes you should create a basin to catch the water. Otherwise, most of the water will be concentrated near the bottom of the slope and the roots may have trouble getting water.

Good luck!

Jeff