The most common place for this sort of short to occur is at the trailer plug, which I presume is the standard 7-pin Bargman type. The Prodigy is sensitive to any current flow , so it doesn't take much to trigger the "Short to Battery" diagnostic. It may take time for the water to soak in enough to cause the short, so spraying doesn't always show it.
This is what Tekonsha lsist as possible causes:
* Brake wire shorted to an external power supply
* Moisture in trailer connector
* Contamination in the trailer connector
* Faulty control
Note that a faulty controller is a possibility, and your comment about the display quality might be an indication of an internal problem. Have you contacted Tekonsha (Cequent) to ask about that?
Contact - Tekonsha
2004 American Tradition; 2014 Buick LaCRosse
Homebase in the Ocala Nat'l Forest near Ocala, FL