Let's say hypothetically, a team scored a touchdown but putting them ahead only 1 point late in the 4th quarter and kicking an extra point would only add an additional point. To take time away from the opposing team, can the holder just spike the ball to run the clock further down or is that a live ball? I know a 2 point conversion would tie the game, but also during this scenario, let's pretend this type of conversion doesn't exist.
Spiking the ball stops the game clock from running in general. Perhaps you're thinking of taking a knee as it's done at the end of games to kill the clock.
However, the clock doesn't run during the extra point attempts (both for kicks and for 2 point conversions). So teams scoring a point cannot kill more time.
You can see references to game clock not running here:
... the clock does not run during or after a conversion attempt in the NFL or NCAA college football.
In American football, the game clock does not run during an extra-point attempt
Yes they can down it or even spike it and as mentioned in the other answer the clock does not run on extra points.
However a note and this has already happened multiple times in NFL games after 2 point conversion was added - in the situation you outlined a coach would be foolish to do anything but have their QB take a knee.
A two point conversion might help them extend to 3 points but is risky... but a kick is just plain dumb. Kicking for 1 point to give yourself a 2 point lead with just a little time left on the clock is useless. If you would get the kick blocked and returned that is 2 points going the other way and you lose.