RMAA should work fine on Linux with Wine, I have successfully used it, and also other audio related software, such as foobar2000, Audio DiffMaker, Cool Edit Pro, Visual Analyser, and others. Make sure that the audio device(s) and API are selected correctly. The default audio devices for Windows applications can be configured with the winecfg utility.
I prefer to use only ALSA on Linux, and actually deleted almost all PulseAudio packages, with the exception of a couple libraries that would break dependencies if removed. So, I cannot comment on how well Wine works with PulseAudio. ALSA 'hw' devices are bit perfect, and 'plughw' (I think those are what Wine uses) should be OK, too, as long as the sample rate is supported by the hardware.
The old Windows MME (waveOut/waveIn) API is emulated well by Wine, and WASAPI seems to work in foobar2000 in "push" mode. With either of these, sound quality should not be affected. The last time I checked, DirectSound was not bit perfect in Wine.
If you get a "perfect" result in RMAA, then you might have used a digital loopback. Perhaps it is a PulseAudio issue ?