When I first hear someone wanting to use a shield, I start out by saying a shield is the first refuge of the incompetent. That's not totally fair as there are legitimate uses for shields, but it sets the tone for the real discussion, which is usually about RF emissions or susceptibility and ultimately about bad grounding that is causing the mess.
A shield should be the last refuge of the competent. Shields also have significant downsides, beyond the obvious cost issue. The incompetent believe in the myth that if you enclose something in a conductive box that RF energy can't get out or come in. That's absolutely not true. A shield can also become an antenna if not designed properly.
Before we can talk about your shield, we first have to go over your grounding strategy carefully. Shields and grounding go together tightly. Explain what exactly the problem is you think the shield will solve, how exactly everything is grounded, what the sources of noise are, etc.
In general, good grounding will do more to reduce RF emissions and susceptibility than a shield. If the grounding is done right, a shield can add some extra attenuation of emissions. If the grounding is done wrong, the shield could become a antenna and make things worse. With a good ground, you generally want the shield enclosing the circuit with as few and small holes as possible, connected to the main circuit ground in exactly one place.
Again, tell us more about your circuit, layout, and problem. Then we can discuss more about the shield if it's still appropriate.
Added after update 2:
It sounds like your primary concern is noise getting onto your analog signal. You currently have 3-5 mV noise on the output of the first amplifier, but you want to get that down to 1 mV. You say this is a transimpedance amplifier, but this is contradicted by your gain of 300k, so we still don't know what your circuit really is.
Where is the input signal coming from? How does it get to the amplifier input? What is its reference and what have you done to insure this reference is clean? The real issue is to make this first amplifier stage as low noise as possible. After that the signal is higher level and lower impedance, so it won't be as susceptible. What are the external noise sources that get onto the input signal? How much noise do you get out of the first stage if you short its input?
High PSRR for amplifiers and voltage regulators is good, but keep in mind that only applies at low frequencies. If you have a particularly sensitive circuit, give it its own linear regulator with the power supply inputs to that regulator filtered. Something like a chip inductor followed by a large ceramic capacitor to ground in front of the regulator is usually good. Maybe even two of these in series. The point is to eliminate the high frequencies on the power supply feed such that the active electronics in the regulator can handle the rest. I would want to see the filters roll off at 10 kHz or below. You also want to keep the unfiltered power supply feeds away from the input signal to avoid capacitive pickup. Guard traces can help.
I don't like the two ground layers. Two ground layers can get you into trouble unless they are stitched together regularly. Again, you are thinking shield when instead you should be thinking carefully about grounding. Visualize all the return currents flowing, and make sure the high frequency components don't flow across the ground plane. Use local sub-ground planes under specific sections that either produce high-frequency noise or are sensitive to such noise. The immediate bypass capacitors go to the local ground net, which is then tied to the global ground net in only one place.
Show the circuit of the first amplifier stage and explain how all the grounds are actually laid out.
Added after update 3:
3-5 mV exists even without the 10K and the C1. Essentially no input to the op-amp. This makes me think that my layout is not perfect.
That tells you the noise is not coming from the photodetector, so you can forget about that for now. The noise is either on the bias voltage for the positive input or is on the ground.
Complete two ground layers connected via several vias.
Again, I don't think this is a good idea for two reasons. First, these two planes need to be regularly stitched together. That's not as easy to do right as it sounds. Second, it sounds like you therefore didn't use sub-ground for critical subsystems. Part of the point of these sub-grounds is to isolate the high frequency loop currents to keep them off the main ground. By attaching each sub-ground to the main ground in only one place, it keeps the high frequency loop currents local, and prevents the subsystem seeing offset voltages between different ground points due to currents on the ground plane.
The 3.3 V supply (also the supply for the op-amps) are filtered via a 2.2 µF tantalum capacitor and a pi network (100 kHz roll over) before the supply to the photodiode (that is, before the 10K resistor).
But you don't show any of that. A tantalum capacitor will have poorer high frequency response and higher ESR than a ceramic capacitor. There is really no reason at all to use a tantalum capacitor at this voltage and capacitance. Also, a capacitor by itself isn't much good without some impedance to work against. You mention a pi network, but none of this is shown on the schematic and you only talk about a single capacitance, so that doesn't add up.
As I also said before, 100 kHz is too high. As I said, I would like to see that 10 kHz or less.
We also have 1/100/10 nF capacitors close to the 10K.
Good, but again, they need some impedance to work against. A ferrite bead chip inductor in series with the supply feed would do that, as I said before.
Op-amp has 1/100/10 nF at supply and bias pins
OK, but once again, these need some impedance to work against. A chip inductor in series would help.
Also, again, where exactly do these capacitors connect to the ground? I suspect you are just punching through to your ground planes. Again, this should all be connected to a local ground net connected to the main ground plane at a single point.
The feedback capacitor and resistor are placed as close as possible to the op-amp.
Good.
All signal traces between the photodiodes and op-amps are minimized; we are talking <2 cm worst case
You have already shown this is not where the noise is coming from.
All the critical deemed signal are placed between two ground layers.
Again, this kind of shielding is only useful if you have a clean ground, which I think you don't. If you don't, all this does is increase the capacitive coupling from the noise on the ground to your signal.
The catch with low cost audio chips like this (or even the more expensive LM1875) is that guaranteed PSR is rather low, around 50dB, while the typical one can be much better (90dB promised in the case of LM1875, see below).
The TDA2030 is actually even less promising with only 40dB guaranteed and 50 typical:
And TDA2003 is even worse with 30/36 dB min/typical:
That's definitely going to be audible if the power source is noisy. (Also beware that these numbers aren't quite directly comparable because ST tests at 100Hz noise and TI at 1kHz...) So... if a linear regulator supply is out of the question (which will totally kill any audible noise from that source in my experience [it gives you some guaranteed 90dB with a LM317/LM337 solution]), try a different TDA2030 chip, assuming you're not having fakes or the CXY knockoffs (or try a LM1875 as it's pretty much a drop-in replacement). Luck of the draw with these things.
Your unspecified "cell phone charger" is almost certainly an SMPS with atrocious noise characteristics (better ones exist, but usually not at this price point). See for example what cheap ones have inside: https://www.youtube.com/watch?v=wi-b9k-0KfE
Also, a linear regulator after an SMPS won't be anywhere near as effective as one after a "classic" (filtered) rectifier. The reason is that the typical linear regulator is very good at eliminating mains hum (actually 2x the mains frequency in a bridge circuit), but its performance gets a lot worse starting around 10kHz... and that's exactly where the SMPS noise usually starts. And the audio amp IC has similar curve, e.g. here's the one from the LM1875:
Summary: the guaranteed fix is better (preferably linear regulated) power supply, which you can build yourself. If that's not an option somehow... try the amp chip lottery. However no amp chip will will magically eliminate the horrible noise put out by a super-low cost SMPS.
Best Answer
The typical signal from a typical full-bridge load cell can be in the microvolt range and sometimes quite possibly as high as several hundred millivolts. Noise can affect all levels but is obviously going to be more problematic when the signal is small.
Screened cable with quad twisted cores is part of the solution to noise pick-up but, it sounds like the interference you are picking up is RF and might be also dealt with by appropriate low pass filters before the amplifier used to interface to the load cell.
If you are using an instrumentation amp, these are commonly affected by RF interference and so there are several well-documented options open to you.