Hi,
If you have shore power then what you are doing is called double conversion. It is not going to make much difference to the cost of power, so I'd ignore it.
If you have an adequate solar system, again I'd ignore the losses, which may amount to about 12% of the wattage that the entertainment devices use.
If you have battery power only and are recharging from a generator it might be prudent to turn the inverter off when it is not in use, and put up with the ten minute download. Even then, the loss is a tiny number.
Use a kill-a-watt meter to find out how much power is being consumed, and make your decision based on that information.