Maybe the task of resampling audio shouldn't be done by the encoder at all. The user should simply ensure that proper audio format comes already prepared from the video source. Any kind of resampling will sooner or later lead do audio quality degradation, either done by this encoder or some other piece of equipment.
Sample rate and audio volume should only specified when using analog Line Input, when using HDMI for audio, re-sampling, voume setting etc. should be disabled imho. This also probably causes too much stress on the encoder's CPU, which may lead to other problems later during operation.
Maybe a single option would be needed for HDMI audio: downmiix to stereo, for the cases when 5.1 or 7.1 audio comes with the HDMI signal, to downmix that to 2.0 audio - but that shouldn't require resampling either.
If HDMI audio is used, the encoder should simply create the encoded live stream at the audio sample rate which is detected at HDMI input, that's all.