I need < 5mA output @-1.22V
If you are only supplying up to 5 mA then there are better options than a negative voltage regulator; consider using a precision shunt regulator. Firstly, it doesn't care whether you are generating +1.22 volts or -1.22 volts; it only cares about regulating the voltage across it.
So, if the ADM8825 is supplying -5 volts\$^1\$ you'll need a series resistance of about 680 Ω to drop the voltage to the shunt regulator. And, if you look into what the dynamic impedance of a shunt regulator might be you'll probably find it can be as low as about 3 Ω in the frequency range 80 kHz to 180 kHz. See the data sheet for the LM4041 for example using a 1 μF output capacitor: -
So now, you can work out what the ripple rejection will be by taking the ratio of 3 Ω to 680 Ω and converting to decibels. More precisely it's this: -
$$20\log_{10}\left(\dfrac{3}{680+3}\right) = 47.14\text{ dB}$$
And, you can improve this by spitting the 680 Ω feed resistor in two and putting a 1 uF capacitor at the junction. This lowers the perturbations of ripple current being handled by the shunt regulator. You might find that at 80 kHz and using a 1 uF capacitor, the ripple is reduced by another 30 dB.
Am I approaching this correctly? What would be the normal design
approach for something like this? Could I practically add my own
filtering to improve ripple rejection?
Well you need to consider shunt references for sure - they are suitable for positive and negative voltage control and are more commonly available than negative voltage regulators. I feel confident this is a better approach and will give you a more repeatable initial tolerance on voltage, lower drift with temperature, lower noise and better long term drift.
The LM4041 is available with an output voltage of 1.225 volts and an initial tolerance of up to 0.1%. The basic device is 1% initial tolerance and this means it will produce a voltage of between 1.2128 volts to 1.2373 volts.
Look at the LT1964 - it has an initial voltage variation of between 1.202 volts and 1.238 with a 1 mA load hence, the shunt reference is better. Looking at the LT1964 it looks like it will drift more with temperature than the LM4041 too.
I'm not trying to press you into using the LM4041; I'm just saying that there are better devices with more options to consider if you think about the shunt reference device.
\$^1\$ I'm assuming you are generating -5 volts because it roughly corresponds with this graph: -