get that you can't get 12v output if you only have 10v input.
Indeed you can't when using a "regulator" but you can with a boost converter.
But in your case (you want 12V output while the input can vary between 10 V to 14 V) what you actually need is a buck/boost module.
Here's an example:
Note the light blue rectangular component, it's a potmeter. It has a screw you can turn and that will adjust the output voltage. You will need a voltmeter or multimeter to measure the output voltage and turn the screw such that you get 12 V at the output.
The 110v AC-12v DC power supply that came with the dog fence provides 12v DC at 750mA.
That means that this power supply can deliver up to 750 mA. The actual current consumption of the device will actually be a lot lower. The only way to know the real value is to measure it.
But assuming it is consuming 750 mA (0.75 A) at 12 V how long will a car battery last? Simple: An average car battery is for example 40 Ah meaning it can deliver 1 A for 40 hours. Or at 750 mA: 40 Ah / 0.75 A = 53 hours. I'm ignoring any losses from the buck/boost module here to keep things simple. We just want a rough indication of the runtime.
In practice I expect that the dog fence unit will consume much less than 750 mA. A more realistic estimate might be 100 mA (just my guess). That would mean that a fully charged 40 Ah car battery will power the fence for at least 2 weeks.
If you keep the battery in your car (and also use it for your car) while it powers the fence I would not run it for longer than a week as you might be unable to start the car if you run down the battery too much!
To fully charging the car battery again you should be driving around for an hour or so at moderate to high speed. I am unsure if the car will charge the battery fast enough when it is running idle.
If you power the dog fence regularly from the car I would get a mains powered car battery charger to charge the car battery instead of burning fuel for that.
Also consider using a solar panel + battery solution instead of the car's battery since the dog fence device should not consume that much power so a solar panel might provide more than enough power.