Sure you can go that way. However it will take a couple of hours to put back the energy into the battery from running the hair dryer for a few minutes. Same with the coffee maker.
It would be better to run a small 2,000 watt Honda generator for such high wattage appliances, then run the TV or other low wattage things with the inverter. But whatever you want. . .
The Honda is about $995 at a place like this.
Mayberrys.com Ask about delivery costs, as that can get expensive. I think this place is free shipping, but don't have current information.
Honda 2000 watt is only about 49 pounds, and you can pair two of them to make 4 KW and run a air conditioner.
If you really want to use a inverter to power the hair dryer, microwave and coffee maker, it is possible. A pair of 105 AH batteries will each hold about 1,200 watts, but only use about 80% of that for long battery life, so about 900 watts can be used from each.
The microwave would require pure sine wave power, while the hair dryer and coffee maker will not. Pure sine wave might be a 1800 watt Prosine, or Outback inverter about the same size.
SunElec.com sells both inverters and solar panels.
Instead of running a noisy generator for a hour or two to make up for the power used, you might like to have a pair of 150 watt solar panels on the roof. My RV uses 35 AH daily to run the CO and propane detectors along with the refrigerator. That is about what one of my 120 watt solar panels can put out in a day. I have a total of 400 watts.
The Honda generators are very quiet, less noise than your truck at idle.
Good luck!
Fred.