I want to start off by apologizing for the very long post
I'm not sure if the TL431 is the best options, its tempco isn't as low as some of the other IC's available in the market. However, the schematic will be similar with all of these. Here's a something basic:
R=(Vsupply_min-2.7)/0.003 - this will allow at least 1.5mA (a little less because of the resistors in parallel) of current through the reference at all times, which ensures it doesn't stop working correctly.
The reference is of a different model, but I've only used it for its symbol, the TL431 could be used instead.
The 1K-10K will ensure the output voltage is >2.684+2u*1K=2.686V which is larger than the worst case reference of the regulators. The trimmer should be of a large value (5K or more) so it doesn't eat too much of the current in this branch of the circuit.
For an fc of 20Hz the cap should be about 22uF - but its optional.
With worst case variations in temperature (assuming we work with a temperature range of about 10-50C in the case of the amplifier) the reference will change by about +-10mV. There are other factors such as the tempco of Iref, the tempco if Iadj (of the regulators), and the tempco of the internal reference of the regulators (which only shows nominal curves in the datasheet). The variations due to current tempco are quite small in this case, so the references are the main issue here. assuming the reference of the regulators has an opposite tempco (which is possible due to variations) we can get as much as 10mA of extra current in the output 1R resistors. The even bigger problem is what will happen if the tempco's are opposite so that the reference IC goes higher and the regulators go lower, which will make the circuit stop working unless sufficient DC current was set with the trimmer.
However, this is indeed a worst case scenario. For 99.9% of times the zero tempco temperature of the reference will actually be in the vicinity of 25C so we should only see a change of 1-2mA at the output with a change of temperature of +-20C. I still think it would be better to replace the regulators for something with a lower minimum load current just to save power, but since that is very easy to do when building the circuit, its of less importance at the moment.
BTW, it is possible to change the circuit so it'll have less resistors and will look like this:
I would advice against this option for our use as it can't generate a reference voltage lower than the IC's internal reference (which could be as high as 2.55V, depending on the exact model of the IC).
Its also possible to implement the voltage divider using a fixed resistor and a trimmer in series. This option is also not as good as it both requires more parts, and uses a divider with parts of different tempco (a constant resistors and the trimmer).
I have another question about this circuit. What happens if the amplifier it drives has a different DC current from each of the supplies? At DC the output caps do nothing, so if there's a constant DC current to/from the virtual GND the circuit might not work. Let assume you have a difference of 10mV's between the 317 and the 337 so a current of 5mA in the 1ohm resistors. Now we add some current (lets say 10mA) from the amplifier to the virtual GND (because there's a DC offset at the amplifiers output, or because someone used a LED returned to the virtual GND, or whatever). Now these 10mA go to the 337 through a 1ohm resistor, dropping 10mV across it. So now the 1ohm resistor at the output of the 317 has 0V across is, so no current is coming out of the 317, and its not operating as it should. This is a big issue with these output resistors since they make the output resistance of this structure very high at low frequencies.
A different direction:
The way we are trying to do this stabilization is actually not the smartest. If you'll think about it, we are trying to regulate the current using an open loop. By open loop I mean we have no feedback of what the actual current flowing through the resistors is. The best option to do this would be by setting the voltage of one regulator at ~Vin/2 using a simpler method such as resistive divider, and than regulating the adjustment pin of the second regulator to to keep a constant DC current at the output. This must only work on frequencies below the audio range.
If we try and solve both these issues (DC current causing the circuit to stop working, and keeping the current from rising significantly with changes in temperature) we should make some changes. This is the first idea I have of how to do this (its more complex than the circuit we were discussing, and I'm sure there are issues with it, but its an idea

):
Keep in mind I didn't test/simulate this, this is just an idea. There's always the chance I've made some stupid mistake and the entire circuit won't work. Also don't pay attention to the part numbers, I just took some parts with the right symbols and didn't pay attention to the exact part number.
The idea of the circuit is quite simple actually:
R1-D1-R5 is a simple divider to create Vsupply/2-1.25V. This is used as the Adj voltage of the 337 to make the output Vin/2. D1 is a 2.5V zener (or 4 diodes), its exact voltage isn't very important, it'll just shift the virtual-GND around the mid-point a bit, it won't affect the current at the output.
Then the 317 is connected with its output directly to this point - but now there's a closed loop used to keep its DC current constant.
R2 is used as a current-shunt to create a voltage proportional to the current of the 317 regulator
R3-C1 is used as a LPF to make sure we only regulate the DC current (very slow changes like that of temperature), and don't affect AC dynamic of the circuit.
Q1-trimmer are used to generate a DC voltage referenced to the positive supply. The jfet is used as a CCS, the trimmer allows setting the DC voltage to the input of the op-amp.
The op-amp should be able to work with its inputs very close to the positive supply (perhaps something similar to the tl072 which is very cheap).
C2-R4 are used for compensation. The low frequency path is through IC4-R2-LPF, and so for stability we must allow a different feedback path at higher frequencies which is accomplished using C2. R4 is there to add a significant resistance between C2 and the supply in case the trimmer is of low resistance, or the amplifier will see a very low impedance load at high frequencies. It has to go the the inverting input while the low frequency feedback goes to the non-inverting because of the phase inversion from the regulator+R2 path (higher voltage at the output of the op-amp generates lower voltage at input of the op-amp).
The resistors R6-R7 can be much smaller now (in theory even omitted altogether), as we directly regulate the DC current of the 317, and the 337 will simply sink any current from the 317+the load.
To solve the issue of DC current from the amplifier all that has to be done is set the trimmer value accordingly. If the current is coming from the amplifier into the circuit, than just set a few mA in the 317, and the 337 will sink the current of the 317+the current coming from the amplifier. If current is flowing from the circuit to the amplifier the trimmer must be set so that the 317 has a few mA more than the current the amplifier draws, this will ensure the 337 is on and sinking current (or to sum it up, set the trimmer so both regulators have at least a couple of mA flowing in them).
The feedback factor B of the closed loop must be considered for stability. Assuming the regulator has a constant voltage between the adjust and the out pins (gain of ~1) we get:
B=1*R2/[(R6+R7)*(1+sC1R3)] at lower frequencies. This means we might indeed have an issue with stability unless we use a very low frequency at the LPF (the lower R6+R7, the lower the BW must be). C2 helps us with stability since B=~1 at higher frequencies, maintaining sufficient phase margin.
The current value is set by the trimmer, and it'll be (Vtrimmer/R2=Ijfet*Rtrimmer/R2) where Vtrimmer is the voltage between the two pins of the trimmer. The exact value of the jfet current isn't critical since we can set the trimmer to compensate for that.
I want to repeat it once more, this is just an idea I've had, and I've only made a very quick stability analysis which at least to me seems to make sense. If anyone could go over it and verify it'll be great, a fresh point of view is always welcome. If anyone would like to prototype it/simulate it, I'll be more than happy to help with calculation of the components values.
I apologize once more for the long post
Edit:
there's a mistake in the schematic, the Adj of the 337 should connect above the zener and not below it.