I think it would be incredibly rude of anyone to pressure your DH (or anyone else who is reluctant) into taking part in something like this.
My response would depend a little bit on the personalities involved. If this is going to be something your DH is going to worry about, I think in fairness to him, you or he need to say to your Mom, now "It's an interesting suggestion, although I'm not sure recoding it when we are all so inexperienced is a good idea - but it's really not my/DH's thing so while he/we will be happy to watch, he/we won't be performing"
And if she tries to push it, something such as "it's not everyone's idea of fun, and it's not kind to try to force people to do things they don't feel comfortable with."
If your DH is not going to be too stressed between now and then, hen I would be inclined to say something now, but keep it vaguer "I'm not sure that that's going to work, given, our musical abilities, and you may find not everyone wants to join in" and then see how things go on the day. But in that case, be ready to back DH up when he declines to participate on the day.
(I have to say, for me, I would have no problem with a sing-song with my immediate family, if any of them wanted to do that. I would absolutely not, under any circumstances, want it to be recorded and sent to people who were not there. Does your DH have a problem with the signing/playing, or with the recording? Because if it is the recording then it may be worth suggesting to your Mom that you stick to enjoying each other's company, and perhaps making some music together, but stick to taking a family photos to send to extended family .)