If the cable lengths are going to be rather short (OP mentioned ~12" or so), then voltage drop won't be a problem.
The Pi will work just fine on 10/100 Ethernet, which uses wires 1, 2, 3 & 6 for transmit & receive.
If you want to wire your own "poor man's POE" as I like to call it, just use the remaining wires, splitting them out on either end. When crimping on the ends, only insert the four wires (1, 2, 3 & 6) into the RJ45 plug before crimping (on either end).
I have seen commercially made (cheap, of course) patch cables that only have four wires in them.
If you want to incorporate a bit of voltage drop insurance, you can always put a voltage regulator at the end by the Pi. Just make sure that the regulator can handle the expected current draw, and that the voltage drop across the wire won't droop below the regulator's minimum operating input voltage at the maximum expected current.
If you have a POE switch, then the best option is either a POE HAT or a POE splitter that drops the voltage down to 5 volts. For regular POE, a 5VDC splitter should be able to provide about 2.4 amps (~12watts) at the far end. If everything is 802.3af or 802.3at compliant, then you should be able to run as far as 100m. You could also use a POE injector on one end (in lieu of a POE switch), and a POE splitter on the other. The overall goal is to minimize voltage drop across the Ethernet cable (which is specified to be able to work up to 100m) by raising the voltage prior to impressing it on the line.
Also keep in mind that you can't just plug a RJ45 plug into a POE switch and expect to measure 48V anywhere. A POE switch won't apply power to the wiring until a test is performed that validates that the end device is POE capable.