Allow current draw only when it is over a certain value
I have made a possible solution. I don't know if this would work for you, but let me explain the principle behind it.
We know your load is 10Ohm (assumption, at 12V it means the load will need 1,2A). The voltage drop across 10Ohm is neglible. But when the load is 1k, the voltage drop is 11,6V-ish.
We set a setpoint with R3 and R4. Where if the voltage after the load is less then 8.5V the comparator will output 5V. The gate of the Pmos will be high, so inactive.
This is a very crude solution, where many unknowns need to be known, but there is potentional for it to possibly work. If we could have more information we could possibly fine-tune this solution or look for something else.