Thanks for the idea! I was eventually able to find the relevant chunks and change the sorting order after some trial and testing.
One thing that threw me off for a while, was the number of "Chunks". Each layer strangely had 80 chunks, many of which were duplicates. The tilemap is 128x128 tiles and each chunk is 32x32, so there should only be 16 chunks per layer. These chunks didn't seem to be used at all by the game. Initially I was grabbing these chunks and changing their mesh renderer's sort properties, and nothing would happen. Eventually I realized that when I started the game, new chunks were created and these "junk" chunks really weren't used at all. Is this a known issue? What causes these unused chunks to show up?
The other issue that concerns me is if I need to worry about refinding chunks? It seems like it would be possible to store the chunks in an array, and then go through them and change their MeshRenderer properties when I need to change their layer order. However, I'm not sure how these chunks are managed. Are new ones continually created so I should go and refind the chunks? Or are they reused?