I agree with the need for an auxiliary start-up power source.
An initial view of an overall power processing strategy for your system needs is as follows: Create an inverter of some sort to chop the source current and drive the primary of a specialized transformer, designed to carry such a current and to operate at the chopping frequency. Probably the chop freq should be chosen to allow a 1-turn primary to be used, since any array of conductors that will carry 2000 amps without disturbing levels of Joule heating is going to be rather thick and resistant to winding.
Once you have that transformer driven, there are many choices, but essentially you will need a rectifier, smoothing filter, and possibly a high-power DC/DC converter to control the quality of the desired 5 volt power.
You need about 100 watts at the load end. Considering the very low voltage of your source, I would not initially assume you'll have an overall efficiency greater than 50%. I expect there will be a lot of loss in the chopper, due to the very low voltage of the source. The source voltage is not large compared to ohmic voltage drop in the chopping MOSFETS. Switching losses might also be large. Assuming you need to get a continuous 200 watts from the source and you will get 2000 amps, you need the source voltage to not fall below 100 mv. However, you describe the source as 10 to 200 mv. Even with perfect power processing efficiency you will need a reliable 50 mv.
Is this source feasible, considering your load requirements?