You should be concerned, but in my opinion not completely terrified of overloading the wiring as some here seem to be. For sure overloaded wiring is a significant fire hazard, but there are protection devices installed (and legally required) to prevent such conditions precisely because it can be a fire hazard.
The wiring is protected by fuses in the DC distribution panel. If it's properly designed and installed (which are not givens with RVs), then the fuse will blow before the current level is hazardous in the wire. (For this reason, don't replace blown fuses with larger ones.)
A typical household box fan uses somewhere around 100W on high; the inverter with that load would be consuming about 10A of current. That should not be a real problem. Note that in this case the inverter size is mostly irrelevant; a 1000W inverter and a 300W inverter would behave more or less similarly. If, on the other hand, you were powering something that required a lot more power (like a microwave or a space heater), then the wiring to the inverter becomes a much more significant concern and the existing wiring would not be at all adequate.
