Nobody in town had a plastic welder, and I ended up not needing it. I just used my soldering iron and the pieces I cut off.
It was actually pretty easy and only took about 20 minutes. Once I got it braced together, I used the soldering iron to melt the two pieces together to "tack" it together in the corners and every 1 or so. After that I melted a furrow at the seam, and then melted the cut off material into the furrow. Then smoothed the seam with the iron as I went.
It worked surprisingly well.