Upon opening the K-A-W I found a different looking board than those shown in the instructions online. Subsequently, I found on the forums here that there seems to be an increasing variety of boards in the K-A-W these days. Mine had the chip tucked away in a particularly hacker-unfriendly way and the blasted chip is tiny. Fortunately, it is still the same chip as called for by T-A-W -- with the same pinouts.
I found one other set of pictures on these forums of this board style (there may be other pics posted but I didn't see them). The pics I did find here were for a board that hadn't been modified yet with Tweet-A-Watt. I am posting pics of mine in case that helps someone. Roughly three years taken off my life soldering onto the teeny-tiny little pins. In the end, my soldering job on this is ugly but it works. Proof positive though that even on these boards it is possible to make the necessary modifications.
To accommodate the wires needing to come around the board I tried cutting one corner off of the PCB and using a Dremel tool to remove where the screw would otherwise go in that corner. Unfortunately, the LCD connectors run all along the top edge of the board and my wires interfered with its function. My second attempt involved very carefully notching the side of the board so that the wires could pass around. The wires do press into the foam on the side of the LCD but with no negative side-effects that I have noticed.
Next time (assuming that I get another K-A-W with the same internals) I will leave the corner of the board alone and just go with the notch in the side.