Careful with those regulated/dimmer 12V LED supplies.
AFAIK they could be PWM controlled, duty rate 12V on/off switching. This works far more efficient for lighting.
Normally the voltage control of the 12V is fairly minimal, doubt they go down to 8.5V.
There are proper DC step down converters for 10A and more but they should also have a display of voltage at least.
Not all will survive the sudden 14V external on their output.
A 10A diode on the output for reverse protection would help but the voltage will not be so stable any more because of their voltage drop is dependant on the current.
In fact you could just try a whole bunch of these diodes(they are very cheap) in forward direction as voltage drop in series to the Auxilary Battery and use that instead of any external power supply source.
At 5 amp they would drop around 1V each, 10 A a little bit higher but not massively like using resistors. I mentioned elsewhere that resistors didn't work for me as the voltage drop varies far too much under different loads while the system powers up.
So diodes should behave more stable, like a deep discharged voltage battery that will also drop it's voltage somewhat with load but NOT like ohms law with resistors.
3-4 in series might work. Maybe a Schottky diode in between for a smaller drop to get a better resolution.
Think of the Schottky like half a standard diode.
So trying out combinations of all these diodes until you get the relay response under 4c of the A-Method.
Diodes must be 10A or higher and if you are holding them in your hand they could get quite hot very quickly. Use at least 1.5mm^2 wires for the battery to terminal inline connection.
A terminal strip should be OK to screw them in series.
ALL AT YOUR OWN RISK
