As others have noted, your battery capacity is at least as important as the inverter size. If you have or can borrow a monitor such as a
Kill A Watt, use it to determine the typical wattage you use per day over a week or so. Use that figure to determine how much battery you need. Generally, you don't want to draw the batteries down below 50%, and you will need to convert the wattage used to amp hours to see what you need.
As to the inverter, you are going to need something at least 20% larger, and preferably 50% than the microwave rating, and a pure sine wave inverter since some microwaves are unhappy with modified sine wave inverters. Best guess without making actual measurements is you will be looking at a 2.5KW to 3KW inverter and at least a pair of 12V deep cycle batteries. You may find it more efficient to use a small inverter to power the TV - generally, the larger the inverter, the higher the efficiency losses.
Of course you also need a way to put the energy used back into the batteries, either a lot of solar, a generator, or hitting a campground with hookups every couple of days.