"Deep Discharge" probably relates to a deep discharge battery - not a cranking battery.
Although deep discharge batteries can be discharged 50% (down to 20 % capacity remaining) according to many specs [even if it only lasts 1 year instead of 3 @ 50% etc], cranking batteries should only be discharged 20% (80% remaining).
Adjust the voltage cutout up by ~0.6%. If it isn't adjustable, insert a diode in line with its sensing terminal. (Almost any diode if its a dedicated sensing wire. Else with whatever current rating that wires has - eg, input power wire.)