Doing some investigating, I discovered that every time it crashes, it's when it's trying to download the 64-bit version of cityofheroes.exe, and it always fails on the permissions stage. For some reason, when it creates the hc-bin64 folder, it gives it permissions that prevent it from writing to it with that specific file.
Oh yes, and deleting the entire folder and re-downloading doesn't fix it any more.
TequilaActivityLog.txt