If you replace the incadescent bulbs with something like an LED string, you wouldn't need the inverter at all. But, yeah, you can run a _small_ (very small!) inverter off of the main trailer wiring. Remember that that power is running a long ways down a little wire.
FWIW, the guage of the wire used for a given load is also determined by the length of the run - the longer the length, the heavier the gauge needs to be to reduce resistance in the wire.