So here's the math. Short answer is that it depends on how long you want to run the microwave.
My Rpod microwave draws 12.3 amps at 115 Vac. That's about 1414 watts. Assume 90% inverter efficiency and get 1571 Wdc or 131 Adc at 12V. Probably it will run more like 11.5V at that current draw so that's 137 Adc.
You will need at least a 2Kw inverter, and it would not be able to run anything else while running the micro. There are two types of inverters available, pure sine and modified sine, which describe the AC voltage waveform the inverter generates. The modified sine wave inverters are much smaller and cheaper but microwaves don't like the modified sine waveform as well. Before deciding which to buy, below is a great youtube video showing what the differences are when running a microwave.
https://www.youtube.com/watch?v=wimTJw_Gpgk
I am assuming a pure sine inverter in then energy calcs below. The modified sine inverter would probably be a little worse.
The 10% inverter losses show up as heat, about 160 watts worth so you will need to mount the inverter in a well ventilated location (but not exposed to weather of course). It should also be as close as possible to your batteries to keep wire losses as low as possible.
I haven't installed an inverter in my 179 yet but my thought is to place it in the forward section of the open area under the bed that is otherwise pretty inaccessible, and install some screening between that and the accessible part of the underbed area.
Regarding energy consumption (power x time - which is what matters for battery and solar sizing) the story is better depending on how long you plan to run the microwave daily. If for example you're only running the microwave for 3 minutes a day to heat some popcorn then you would only consume about 7 amp hours daily. That would be about 3.5% of a typical 200 AH dual battery system capacity and 20 watts of solar could replace the lost energy on a sunny day. Not a big deal, and it eliminates the annoyance of having to start the genny just to run the microwave for a few minutes.
If OTOH you want to run the microwave for an hour a day doing some real cooking then you would use 137 AH which is more than half your total battery capacity and you would need over 400 watts of solar to replace the energy used on a sunny day. In that case I agree that just running a generator would be a better solution, although these days you can probably buy the solar, a charge controller, and the inverter for about the cost of a good 3kw genny.
I'd suggest starting with the genny if you already have one or camping with hookups first to see what your real world typical microwave usage is in minutes per day before deciding on the inverter. If you change to a 12V TV and use 12V usb or cigarette lighter power points for phones and pads, then the microwave and air conditioner are really your only remaining AC loads (unless of course a family member insists on using a hair dryer while boondocking

).
The air conditioner is an even larger load than the microwave (the horribly noisy and inefficient 13500 btu/hr roof air on my rPod takes 15.3 amps AC) and needs to run for long periods, so while you could get an inverter to start and run it, its not really practical to run on batteries and solar. I don't know why RV air conditioners are so much worse that even a cheap window shaker. I'm looking into converting to a high efficiency mini split (which are close to double the efficiency of the RV roof units, and quiet too) and mounting the condenser on the tongue where the batteries are now. With the roof space that frees up for solar and a lithium battery pack it should be possible to run the air conditioner at least part of the day while boondocking.
In the meantime if anyone is interested in buying a litte used rPod roof air with an Easy Start already installed so it can run on a 2kW genny I might have one for sale soon
