The problem with running a 12-volt power supply unit from a 12-volt battery is that your output voltage cannot be greater than the input voltage unless the input voltage oscillates sufficiently to run a step-up transformer/circuit.
Step-up circuits are very sophisticated and there is absolutely nothing inside a computer's power supply that can be modified to do this without cannibalizing the unit--rendering useless its function as a power supply.
The input voltages to the devices inside the computer are all 12-volt and down. What you need is something that will step down the car battery's voltage to the voltages required by the main board. Your attempt at doing it by modifying the power supply's circuitry indicates to me that you have no technical expertise at such a task or you would know that it cannot be done.
What makes the power supply work is the ocillating current from the 110-volt wall outlet. You cannot get that from a flat signal (battery) without an inverter (oscillating step-up circuit)
There is a way to do this with a battery and no ocillation by using voltage regulator ICs; but, you would not be able to run it for long because the battery would very soon drop below 12 volts--unless it were being constantly charged.
If it were to be accomplished, it would be better done by starting with two, twelve-volt batteries wired in series for twenty-four volts total. That way, it would be quite some time before the batteries dropped below twelve volts total.
Using this method, you could then simply set up several voltage regulator ICs (integrated circuit transistors) to put out the desired voltage. These are easy circuits to build; but you will need several of them. It has been quite some time since I built such circuits; but, if by memory serves me right, you will need several TO-3 power transistors and maybe a few TO-220's. You'll also need a conglomeration of low-tolerance resistors and maybe a few POTs.
If you don't know what these things are nor how to make them work, teaching you here would be too large a task. It requires an education, not a how-to sheet.