I considered asking around about a welder but the cost will exceed a replacement. I can wait for a new one, assuming someone on earth sells them.

I went ahead and pulled the axle out. I set my calipers to 0.37" (3/8"). They slip over the smooth section with room to spare but won't fit over the threaded section. In metric I'm getting 9.2mm smooth and 9.8mm on the threads. So basically the axle doesn't match any of the expected numbers (0.375" or 9.5mm), unless I'm measuring wrong, using the flat part of the caliper.

I think I'm getting a thread pitch just above 1.0mm, so my guess is that it's 26tpi.

I'll try to do pictures.