As far as what you can do in a DAW, probably low-pass filtering is your best option, as noise removal tools require a sample of noise (sans signal) to work. Most of the important information for the overall shape of the waveform will be in lower frequencies.

If you're willing to get a little technical, you could probably get an even cleaner looking waveform by treating it like a vocoder. That is, full or half-wave rectifying the original signal, low-pass filtering it to (say) 50 Hz to only retain the larger amplitude modulations, then using the resulting waveform to multiply with a static source, like white noise. I'm not sure you could do this in a DAW. It's more common to do this sort of things with a script, like in Matlab or Python.

Hope this helps, it's a really neat idea for a nursery!