Generating a dirty sine wave which will power an appliance is fine, but that's not the goal here.
The goal is to generate a clean sine wave which will benefit the grid when applied to it. This is probably impossible at bike workloads, and is in any case quite a bit more of a problem than just throwing an inverter at a DC source and calling it done.
200W isn't a lot of power. Most microwaves are 1KW. You'll need an inverter that can do that at least to run a fridge constantly and a microwave periodically.
The goal is to generate a clean sine wave which will benefit the grid when applied to it. This is probably impossible at bike workloads, and is in any case quite a bit more of a problem than just throwing an inverter at a DC source and calling it done.