If a high frequency signal is passing through a capacitor, does it matter if the capacitor is charged?

The expression "capacitance changes depending on DC bias" is a bit misleading. It actually comes from the fact it is tested with a DC bias and a tiny AC voltage added to it to measure the capacitance. But in reality, the capacitance of any capacitor always depends on the instantaneous voltage across the capacitor, no matter where this voltage came from. Usually it is DC, but in your case it will be low frequency AC plus high frequency AC.

So you need a cap with low \$ dC/dV \$ or capacitance variation per unit of voltage variation. That includes C0G ceramics and most types of film caps like PPS, PP, etc.

In your case, since mains voltage is involved, be sure to pick a capacitor rated for it. Not just the AC voltage, but it should also be rated X. This is not about movies, rather it means it is safe to use the capacitor across AC mains voltage. Mostly this is about not starting a fire if there is an internal short in the capacitor, so it will have features like self-healing. Note the capacitance value of these tends to decrease over time as they are exposed to voltage spikes which cause internal arcing, which pokes a hole in the dielectric and causes a short. At this point the self-healing feature kicks in and the metal around the hole vaporizes, which means there is no longer a short, and prevents fire. But it also means the capacitance value has decreased a bit. So if you want long term accuracy, maybe pick a higher voltage rated cap, like 600V or 1000V.

Also you should put high value resistors across the caps to discharge them once the device is unplugged, because I guess this device is going to have a male mains plug at the end, and it is never a happy experience to grab one of these when there is a charged capacitor on the other end. The resistors will also balance voltage across the caps so they both have half the mains voltage, so they're both as far as possible from their max voltage rating.

Several other people mentioned Class 2 ceramics are the absolute worst, but due to having strong dC/dV not only will the capacitance be modulated by the 50Hz AC voltage, it will also be modulated by your HF signal which will create all sorts of distortion harmonics. They are also quite lossy, so some of your signal will end up as losses. They are awesome for decoupling power supplies due to high capacitance per volume, low cost, low inductance, but they are terrible for everything else.

Actually it strongly depends on the capacitor itself. E.g. many ceramic capacitors are known for their charge dependency. Just as a starting point: https://www.murata.com/en-global/support/faqs/products/capacitor/ceramiccapacitor/char/0005 .

So always keep in mind that the given capacitance is often only valid for small signals and can strongly deviate.

Yes, it matters, for real world capacitors the effective capacitance is a function of the voltage applied.

The strength of this effect is dependent on capacitor technology, it is most pronounced in ceramic capacitors with class 2 dielectrics, aka MLCC , the effect can be as strong as -70% effective capacitance when at 100% of rated voltage for smaller sized ceramic capacitors and low grade dielectrics .

The mechanism of this effect is piezoelectric (electrostatic) in nature, the ceramic expands under applied electric fields.