The trailer harness is pretty small gage wire, maybe 16 ga or so, and it is a long run of wire between the alternator/battery and the trailer plug. Wire that small won't move enough current for inflating tires very fast, or you may get the wire quite hot and possibly blow fuses.
If you want a quick-plug 12V compressor as opposed to using alligator clips on the battery, you'd need to wire a dedicated circuit directly off of the battery (ideally with a circuit breaker sized for the compressor). I use something similar for powering the fridge in our camper when towing - 10 ga wire behind a circuit breaker, then a Anderson connector with weatherproof boots for quick connect/disconnect at the back of the rig. It's not hard to wire something like this up, including cutting the leads off of the compressor and replacing them with an Anderson connector. 10 ga should handle 30-40 amps over a short run (i.e., compressor at the front of the rig), which should be fine for most smaller 12V compressors. I do run my ARB twin off of 6 gauge though, as I had my Smittybilt on 10 ga and it would get the wiring a bit warm when running.
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