How do I implement a feedback to keep the DC gain at zero for this conceptual passive filter?

Your filter has an output impedance. The load impedance interacts with that output impedance to create a voltage divider.

If you want to eliminate that dependence, then you need a simple voltage follower (buffer). Connect the "nominal" load impedance to the filter, then use a VCVS as an "ideal buffer", controlled by the voltage across that load. The output will be independent of whatever load you put at the VCVS output.

In the real world, use an opamp voltage follower (unity gain).

Simply put, passive filters interact with loads and sources (as you have found). Adding feedback makes it (overall) an active filter.

The suggestion of putting a buffer amplifier on the output is probably a good one. It'll be easy at frequencies (like the 0.1Hz of your filter) where you can use op-amps, far harder at microwave frequencies.