The problem here is that the 12V output of the ATX supply in question almost always have output filter caps that are rated no higher than 16V, you would be running near the edge of over voltage on the caps..but just the edge. It is always desired in design of equipment in he holy name of safety and reliability to choose components, especially electrolytic capacitors to run at working voltages some reasonable fraction of their max ratings.
It is certainly sane and cheap enough to find two or three 25V caps to replace the 16V ones if you want to try this.
You must also be aware that the 3.3V output caps may also be rated at 4.3V and may need to be changed. The 5-V supply, on the other hand will have 10V or 16V caps so they needn't be bothered with, but it wouldn't harm to peek at their dresses to verify this.
The output voltage is set by use of an optocoupler (4 legs) and it can be found bridging the large cap between the output low-voltage section and the AC mains section. It is certainly possible, if not easy enough, by maybe just adding a 4.7 zener diode or else modifying the feedback resistor values on the 12V side there to set the 12V output to 16V, but don't try going over that. Often there is a LM431 that sets the feedback voltage and drives the optocoupler and there are two resistors on this device that control the feedback voltage.
Some expert knowledge about electronics needed(a knowledge of ohms law, zeners, voltage dividers, how to use a potentiometer to change a voltage setting resistor's value, knowledge about ooptocouplers and LM431 operation.)
But there are plenty of people who will discourage you on this forum to not to try doing this rather than offer advice on how to do it safely.
Wear safety glasses when testing. Capacitors can shoot off the PCB like little rockets when subjected to over voltage.
..batteries not included, some assembly required.