I'm an idiot. I tipped the cabinet on its side to put the last coats of paint on (latex dries better horizontally, I think) and I noticed that when I measured the base I didn't account for the overlap from the side panels so it is actually an inch taller than I thought.
I had to remake the entire thing using a 1 x 4 (3/4" x 3-1/2"). It was easy enough and only took about an hour but I did have to spend an extra $16 on the maple and I still have to paint it black.
The good news is that I finally figured out how to paint the sides without leaving roller marks - I bought a 1/4" nap roller and that did the trick. The foam roller was spreading the paint too thin and it was drying before I had a chance to even it out. It looks real good now.