After a few weeks of having a broken laser, I received my replacement laser power supply and installed it last night. I emailed a few sellers on eBay that sold similar looking laser engravers with pictures of the fried parts, and I was able to fine someone that happened to have a spare and ships from the US.
I was a bit worried about screwing something up, the red power lead you see coming out of the transformer is soldered directly to the laser tube’s terminal. Connecting the new lead was a bit cumbersome with limited space, but not difficult. The new supply fit perfectly in the old place and I did not have to modify any mounting holes or even figure out what unmarked wires served what purpose since all the connection points were exactly the same.
All in all the replacement took about an hour and the laser just worked with no debugging or hassle with the new part.
The biggest worry I had when I originally bought this laser was that if anything broke down I would not be able to find replacement parts. While you probably can’t find any parts locally, there seems to be plenty of places online that can get the parts.