Power Arduino w/ car battery which will sometimes sag while effectively shorted

I would like to make a car battery-based spot welder using an Arduino Nano to control the time of the pulse of a large relay (turned on using a BC337 transistor) and display the pulse duration setting on a seven-segment display. The Arduino would be receiving its power from the car battery which at full chage would be 14.4 V. The thing is, when the car battery is shorted, I expect the voltage to drop and I don't know how much. Would this be likely to affect the running of the Arduino? Would a capacitor be needed in parallel with the battery (I think it would also need a diode to stop the current flowing out of the capacitor away from the Arduino when the pulse happens). If necessary, how large a capacitor would it need to be?

I don't expect the voltage of the battery to sag below the voltage needed by the Arduino Vin; I just expect it will drop suddenly and have no idea how much that will affect the output of the regulator. I don't yet know how long the pulses will need to be but definitely less than 300 ms.

Thanks for any help.

What do you think will happen as the relay contacts close and open?

The same as happens on all the ones made by other people before me. I’m not making this for industrial use and there will be a fuse in case the relay sticks.