I agree that 'musique d'ambiance' can be used. As for Muzak, since this is the trade name of the corporation which started 'piping' music into establishments subscribing to it, it could also be known as Muzak. It is still called Muzak in the UK.
'musique d'ambiance' seems fine to me. Just to say that I've already heard 'musique d'ascenseur' for the kind of bad and unexciting music you can hear in public places like an elevator. It is quite derogatory. I don't know if 'elevator music' conveys the same meaning?
I agree with Edwin, and it is also listed in categories in French as:
Musique/style de musique 'easy listening', which is different as Ambient Music, the latest as well remains musique: genre Ambient. in French. (je sais, je sais, Bastoune va m'arracher les cheveux...)
There were a sort of confusion over the 'musique d'ambiance' for some reason between these two different styles.