RV ac units can often be too much for a regular 110/120 household outlet. Especially if something else is pulling from the same fuse at the same time. Could be happening with the fridge. Try hooking up the fridge to a dedicated fuse with nothing else attached to that fuse (or turned off) - does it still blow the fuse? If so, there's a problem. If not, it could just be that you had to much pulling at that moment.
Does the inverter actually say it's an inverter? It could actually be the converter that turns shore power into DC power. Few older RVs came with an inverter - they weren't feasable in those days. But they most all came with a converter.
And remember that on battery power, your outlets won't work (unless you actually have an inverter), but the lights should work just fine. On shore power, the outlets do work (pull directly from the 110 outlet) and the lights work by pulling power converted to DC by the the converter.
