FWIW, here are a few comments.....
200ah of battery may be bit marginal for a 2000W inverter...not so much for run time, but a pair of group 31 batteries may struggle providing enough instantaneous current to keep the inverter happy with a 2000W load. This can result in a voltage sag down in the 11.X volt range which will cause many inverters to shut down (low voltage cutoff).
Short fat battery cables from the battery bank to the inverter will help the voltage sag.
I have a 700W microwave and use it alot....they typically pull around 1000W of AC power. That being said, my microwave cooks happily with a 1000W inverter.
I have a true sine wave unit that I picked up used on CL but we have another MSW unit and the neighbors have a 2000W Tripplite MSW unit and they power everything just fine so far.
A few other considerations.....larger inverters burn more power idling and inverting, so it's a bit of a waste to turn on your 2000W inverter to charge your laptop for example. For my use case, I just turn on the inverter when I want to run the micro and then turn it off. When running, the microwave/inverter is easily pulling my fully charged 350ah worth of house batteries down to 12.3-12.4V. If you a planning on cooking a Turkey in the microwave, forget it.....you wont have enough battery..
You may be finding that most of the lower priced inverter chargers are MSW. There are plenty of 1000, 1500 and 2000W units around, and sub $1000 should be easy.
Here is a Samlex 1500W MSW inverter charger For $316 and free shipping for example.....
http://www.amazon.com/Samlex-SAM-150...verter+charger
Another.....this one is a pure sine wave inverter charger
http://www.amazon.com/Samlex-TN-1500...verter+charger
As for TV watching....AC will be very inefficient...12V would be a lot better. If it were me I'd use a laptop or tablet or similar......very energy efficient easy to charge portable, less of a theft target, etc.