Does an inverter only draw power from a battery as-needed?
If two 100% efficient inverters, one 500W max throughput, one 1000W, are used to drive the same load, they will pull the same power from the battery.

Of course neither will have 100% efficiency. Even at no output load, they will draw some power. The chances are high that the 1000W inverter would draw more idle power than the 500W, but it's not a foregone conclusion. A well designed 1000W inverter could be more frugal at no and low loads than a badly designed 500W.

For your application, rather than suffer the inefficiencies of two stages of conversion, it will be more efficient overall to use a single 12v input DC to DC laptop SMPS, and one to your router/modem, unless they are 12v input (mine are) in which case run them straight form the battery.