Gain problem of inverting opamp
The LM358 has an input offset voltage error of typically 2 mV. Given that the input is only 5 mV, 2 mV represents a large error signal. On this basis alone, if I needed DC accuracy I would be choosing an op-amp with a significantly better input offset voltage.

Input offset currents are also likely to add to errors.

Adding the capacitor means that the DC gain of the 2nd stage is zero and no further amplification of offset voltages is produced. This may be the effect you are seeing. Try bypassing C1, shorting the two inputs and looking at the offset voltages on the output you get in this scenario.