I think Elvis's laughter is inspired by the twin factors of having forgotton the lyrics, hence his improvisation, and the fact that he is choc full of drugs, which made him not only forget the lyrics, but find the whole thing funny.
The backing singers would have carried on because that is what they were instructed and trained to do - in the hope that keping things going would give Elvis a point at which he could pick up the song again.
When you are as stoned as Elvis ws during his 'Vegas' period, nothing at all is required to set you off - everything seems funny.