"the systems voltage will cave down below 12.7 volts and the relay will automatically disconnect. The voltage then rises, the relay re-engages: then the voltage drops, the relay disengages and a buzzer-like sound will be heard as the relay quickly switches in and out."
What the hey?
This is so funny it is hysterical...
Anyone with any sense designing an electronic switching circuit knows from basic electronics course that including HYSTERESIS as a quality is mandatory. Turning a power relay into a "buzzer" is a sure recipe for early failure never mind causing schizophrenia.
Let me do one of my infamous parody's....
"Please you notice we at Promiscuous Butterfly television works ltd. make for you special feature. When switch channels, both stay on for 10 seconds make sure you miss nothing..."
I cannot believe such junk as what you have described are on the market. With my LED and piezo annunciator warning systems THIRTY YEARS AGO I had to build meaning add a circuit with hysteresis to avoid exactly what you describe.
Using an inner tube stretched between two trees I wonder how far in hundreds of meters I could hurl your device. Preferably onto a freeway where a dozen 18 wheelers could have their way with it.
Seriously. Get rid of this nightmare - fast. Replace it. Sure power has good battery separators. God I wish Lisle Bruney a fellow engineer (Sure Power) was around to read your post.
This is not your fault. You got stuck with a lemon. Heck even a lemon is useable, what you have should go on display somewhere.