A 2000w inverter has the potential to draw more power, but it depends on how you use it. The quality of the inverter will also have a HUGE impact on battery consumption. A low cost (not efficient) inverter will save you some $$ now, but will cost you amp hours.
I just installed the
GoPower IC2000 and think that would be a great solution for you, seeing how you haven't purchased the inverter and transfer switch yet.
I LOVE this thing!! They claim you can also run 1 Air Conditioning unit using the IC2000 AND your Honda generator. I have the small Yamaha EF2000 generator and can't wait to try it.
You can watch my YouTube video about the GoPower IC2000 device here. Installing this will take you less than 1/2 the time it would take to install an inverter and separate transfer switch (trust me...I just did both installs)
https://www.youtube.com/watch?v=jjevQ10M97Y
I also did a transfer switch video for our blog (before I did the IC2000), you can watch that video here.
https://www.youtube.com/watch?v=zTjOECEhYDM