A few weeks ago I taught myself slip-stitch mosaic knitting to make a booga bag (in the round). So yes, you can definitely convert that hat to circulars. It's too late tonight for me to figure it out, but if you can wait until tomorrow, I'll give a more detailed description. I put two of my kids and four daycare kids on the bus at 8:45am (Eastern time), so I'll try to start soon after that! PM me if you are interested. If you've gotten so excited that you figured it out, let me know so I'll save myself some work!
There are a couple of problems with converting the pattern to the round: on flat knitting, two rows are worked identically, which is why the design has "stripes" of two stitches high and wide. On circular knitting, you'd have to slip stitches two rows to get the exact design. This didn't work well for me -- my slipped stitches were huge and ugly. So, I cheated and alternated colors every row, but that meant slipping more than 3 stitches at a time in a row (not recommended, but since my bag was felted it didn't matter) On a hat, it's a different story.
I'm going to post pictures of my bag so you can visualize what I mean.
Meanwhile, I designed a new chart, but before giving you all the directions, I want to make sure you like it. Other options would be to make the design almost identical to the one pictured (but with skinnier "stripes") but then the band would only be half as tall (11 rows instead of 22) You'd just have to make the solid part longer.
Let me know if you like the one I drew, and if so I'll continue on...