Is it 12.26V straight out of the power supply or was that measured on the PCB (i.e. at a pin marked “12V”)? Did you measure this while the LCD was connected? If it’s connected, this will draw current and affect the voltage. I can’t find an order for your MIDIbox username, so I don’t know if you’re using a high-power or low-power LCD.
Setting to 9V might not work, if you had measured 12.26V going into the 9V regulator, then setting the power supply to 9V might drop it by 3V and it won’t be enough for the 9V regulator, which needs 10.5V or more to operate.
You don’t need 12V for 6582A… but you need enough for the 9V regulator. Also, high-power LCD backlights are powered by the input voltage, before going into the 9V regulator.
The issue is, unregulated power supplies should deliver their rated voltage when the current load is at the rated current. When the current load is smaller, the output voltage will be higher. So it’s probably a good idea to find a power supply with a smaller maximum current (perhaps 500mA). However, I’ve run my sammichSID with a regulated 12V power supply, and I don’t think the PCB gets too hot, it’s definitely hot, but I’ve noted that the regulators are dissipating well… if they were getting too hot, then the ends of the heatsink fins would also be getting too hot, which they aren’t.