I tried putting a cap in parallel with the output. Conveniently, you can attach caps to ground via the AV1 OUT jack with the switch in the out position.
From the schematics the output impedance is about 1040Ω, so a 6.8nF cap puts the cutoff frequency at 22.5kHz, which shouldn’t have a noticeable effect on audio quality. This helps a lot, but it’s still fairly noisy. Increasing the cap would help a bit, but at some point it’s going to start cutting in to the audible range.
Ultimately the filtering problem will never be adequately solved with an RC filter; the difference between the passband (20kHz) and the noise/stopband (50kHz) is too small to get good attenuation. An active filter is required, which means op-amps, power, etc. Messy.