I found the solution while typing this. Config information was being stored in C:\Users\NAME\.EjsConsole.txt, which stores the JVM flags entered in the Advanced Options tab. I had to modify the flags to make EJS stop crashing on launch.
I'm going to leave this here, because I think it would be a good idea if this fact were added to the EJS documentation somewhere. I lost more than 3 hour to trying to fix this.
Yes. EJS saves user information in two (hidden) files in your user directory, .EjsConsole.txt and .Ejs.txt.
The first one contains the last configuration you used in the EJS console. The second one signals the path to your last EJS run and the source directory of your EJS workspace. This is used when you try to open the EJS model from a self-standing JAR file.