The Melodie Nelson bass line is one of those unicorns isn't it? It's also one of those rare moments where the sound is so famous, it's a shorthand. Say Melodie Nelson bass and many people know exactly what you mean.
A few years ago, I was in the studio and a song had a perfect opportunity for one. I couldn't ever sound like Dave Richmond did in 1969 or whenever, but we wanted to channel the vibe. We used a Fender Jaguar bass strung with oldish flats and played with a pick and some palm muting plugged directly (no fx for once) into a 1959 Fender Bassman modern reissue and turned it up to the edge of distortion. The rest was the bass part and the engineer's fingers - doing whatever they do. It was one of the loudest instruments in the final mix too - I think that's also one of the important things.
Not nearly as special as Richmond, but the feeling and reference were clearly there. People in the know said "Melodie Nelson bass" when they heard the song.