What are the voltage requirements of the mini PC after the power supply? Is it like most computers with 12v, 5v and 3.3v, or is it a single supply voltage to the computer board? Are there control signals from the PC to the power supply?
The reason I ask is because this may be something you can do with a battery and a series diode instead of buying something pre-made. For example, if the mini PC only requires a single 12v source, maybe you could use a 12.65v lead acid battery with a series diode dropping it to 12v. As long as the power supply is supplying power, the battery is disconnected as the diode will not be forward biased, but once the supply cuts out, the diode will immediately conduct allowing the battery to take over. The PC would see a slight drop in voltage, but it shouldn't be enough to cause issues (in this example).