This is a difficult problem, so you want to do everything possible to maximize the sensitivity. I a building something similar to what you need.
Opamps usually work better with dual supplies, a single supply often has compromises.
The circuit will use really large value resistors, so you want an opamp with really low input bias current. You also want low input offset voltage. After a short search, I found a LMC6001, it seems to meet the needs, although the offset voltage is higher than I would like. Adding a second amplifier stage will only amplifify the offset, so it is not wise. Some of the voltage offset can be nulled with software, although it will vary some with temperature. This opamp has a absolute maximum supply voltage rating of 16V (V+ minus V-), you want to be a few volts below 16V.
You want a guard ring around the sensitive net. The guard ring should be at the same potential as the sensitive net (lowers risk of leakage current). For a dual supply this is easy, since the sensitive net is at 0 volts, the ground fill will be the guard ring. Leave extra space around the sensitive net to reduce the chance of leakage.
You want to avoid wires to the photodiode. You can mount the photodiode on the back of the board so it can be flush with your box.
If shielding is necessary on the top of the board, I will use metal standoffs and a thin aluminum sheet ("flashing" from a home improvement store).
Large value resistors have inherent noise that can be an issue ( https://en.wikipedia.org/wiki/Johnson%E2%80%93Nyquist_noise ), so it is good that you don't need high bandwidth.
There are some extra features that you may not need. I have a small regulator to power my low-power TI MSP430 MCU. I also have a dim LED power indicator (yes, modern LEDs work at < 0.1 mA). If it affects the photodiode, I will disable it, or cover it.
I will run it from one 9-volt and one AA battery. The expected current draw is a few milliamps.
The parts are on order, they will be here in a few days. I just completed the PWB design, I will be ordering it soon.
I will update the answer after I build and test the circuit.
Edit: I built the PWB with the LMC6001. I doesn't work properly, the opamp may be bad or maybe I damaged it. They are expensive, so I only have one. I will order another.
I substituted a TLV2760. This is a low-voltage opamp, max supply voltage of only 4 V (I used 3.3 V, no negative supply, it is rail-to-rail). It has a higher bias current, so I changed R3 to 10 M, and C7 to 0.01 uF. Now, it works as expected. The output still has considerable noise, so a large cap at C8 will be necessary (I will use a 4.7 uF ceramic).
Here is a summary of the key issues to building a high-gain transimpedance amplifier:
You need an ultra-low input/bias current opamp. It needs to be much lower than the current that you are attempting to measure. The bias current of the TLE2072CP is not low enough to meet your goal.
Any offset voltage error of the opamp will show up in the output. Adding a second stage will amplify this error, that is why you want to do most or all of the amplification in the first stage.
Large resistors have a lot of Johnson noise, you want a large feedback capacitor to help filter it out.
The PWB layout is critical. You need to keep the connections short and avoid leakage paths.
The 100 M resistor that I ordered is physically larger than I expected, I used the wrong footprint.
