Paul,
There is a quick check you can perform that will either help you to identify a problem or give you confidence that your inverter is performing correctly.
First of all, purchase a digital voltmeter. It doesn't have to be a expensive model with a lot of features, the least expensive DVM from Home Depot, Lowes, or even the free one from Harbor Freight Tools will do the trick.
Secondly, use the DVM to measure the voltage across the battery terminals with everything turned OFF - it should measure somewhere between 12.1 to 12.6 Vdc depending upon the state of charge and the overall condition (age) of the battery.
Third, turn on the generator and give it a couple of minutes for power transfer to kick in, then remeasure the voltage across the battery terminals. At the very least, it should measure a voltage that is higher than you measured with everything turned OFF, perhaps as much as 2 volts higher even. Shut down the generator as soon as you have completed your voltage measurement so that you don't affect the charge state. Remeasure the voltage afterwards and verify it is close to the original (i.e., everything OFF) voltage.
Fourth, hook up the AC shore power connection to the RV and activate any switches you would normally switch ON in order to charge the battery. Remeasure the voltage across the battery terminals and compare that value to the voltage measured when the generator was running - if they're roughly the same, then the battery should recharge in roughly the same amount of time regardless of which method is being used - genny or shore power (inverter). Disconnect the AC shore power.
Finally, start the vehicle engine and measure the voltage once again. Compare this measurement to that of the generator and the inverter charging voltages - typically, the engine alternator will provide the highest charging voltage... maybe somewhere in the 14.0 to 14.5 VDC range.
Having done all of this, you should now have a fairly good idea of which electrical subsystems are functioning correctly and the relative charging capabilities of each. Not bad for less than a $10 investment and 30 minutes of your time.
Dale