Can you safely power a device with a PoE splitter that's the same voltage but at a higher current & power?

I have a device I want to power via PoE. The adapter that came with it lists +12vdc, 0.5A, 6.0W. Barrel connector 2.5/5.5 mm.

To power it with a PoE splitter, I know getting the voltage is important but what about the amperage and wattage?

Ideally, I would find a PoE splitter that matches it perfectly and I'd call it a day but I'm having a hard time finding one.

Will it break the unit to use something that has the adequate voltage (12vdc) but additional current and power specs? IE 12vdc, 1.2A max, 15.4 W.

If I remember correctly, the amperage and wattage specs are just maximums and the device itself will pull whatever it requires which should be less than its maximum capacity.