Good news! With the help of HD Security Store, the vendor that sold me the Dahua camera, I was able to get the firmware updated. Here is what was needed:
In the config tool, when you select your particular camera and select the "Login" button, you're presented with IP Address, User Name, Password, and Port. By default, the port is populated with 37777. Simply change this to 3800, and Voila! The firmware installs correctly.
I don't know why this works. My camera itself has the TCP port set to 37777, but who cares why it works as long as it does.
I now have the General_IPC-HX5(4)XXX_Eng_N_Stream3_V2.420.0000.0.R.20140419 firmware running on my cameras. Nice!
If you order a Dahua on Amazon, be sure to choose HD Security Store as the seller. Their support makes it well worth it!