Increasing sensitivity of photodiode amplifier
For detecting very low radiance sources an integrating TIA is the way to go. Just remove the feedback resistor and size the capacitor to set the gain of the first stage. Add a switch in parallel with the capacitor to reset. Not seeing much instantaneous current isn't surprising for small signals and is exactly the problem that an integrating amplifier can solve.
You should put some time into calculating the expected irradiance on your detector and use this to determine how much current you get. This will allow you to properly size the feedback capacitor and better determine how long of an integration you need. It sounds like you are feeding this into an ADC. You will want to select the capacitance and integration time such that your capacitor fills up to somewhere near the maximum input voltage of the ADC for the highest irradiance you expect. Doing so usually improves SNR and otherwise you will be wasting some of your ADC bits and in the extreme case can add significant quantization noise, further reducing SNR.
Once assembled, you should be able to watch the output voltage ramp up during an integration with a higher slope under illumination than in the dark. If your signal source is comparable to the dark current of your diode you will need to subtract the dark current contribution after you complete an integration.
As was pointed out in the comments, your second stage amplifier is another TIA, and shouldn't work as drawn. You probably can do what you want with just one stage, which is usually best for noise performance. You also should be able to set the reverse bias of your photodiode as 0 V can work, but isn't going to be as good as a bit of reverse bias.