Hmm Sharon, I must say I don't envy your position.

I would tend to agree with the "Only sing along if invited group." Especially if the audience drowns out the performer in some cases - that's a problem. I've also seen instances such as you describe where the performer is trying to do a version other than the usual one and is consistently thrown off by the audience who's singing it like its "supposed to be" sung, as a matter of fact that's happened to me. I tend to only join in on choruses if invited and if they really are singable - like sea shanty type choruses. Otherwise I prefer to just listen closely to the singer's take on the song, without joining in. I find it excruciating when the performer insists repeatedly that the audience join in, to the point of treating them like kindergarteners. As in "Let's try that again, I couldnt' hear you...".   A simple invitation to join in should be enough, and if the audience doesn't want to then so be it. In the case of your club it sounds like you have some very enthusiastic singers!

In your case you'll have to be extremely diplomatic to avoid upsetting either group - the joiners in and the non joiners. Good luck!