I always found this logic as to why they cut the song out to be absurd. The Muppets are meant to be for children AND adults. Hell, the original Muppet Show from the 70's was targeted towards adults.
The song is on my dvd it's actually the only scene in the entire movie I do not care for. I always fast forward I love the songs in this movie otherwise and the muppet christmas carol will always be my favorite
I am guessing that you are very young and have not yet experienced loss in a romantic relationship. For me, Michael Caine's acting during that song speaks volumes and gives a deeper understanding for Scrooge's coldness than any other version of the story. Without it, they may as well have cut out any reference to Belle's existence.
When I was in grade school, they asked me what I wanted to be when I grew up. I wrote down HAPPY.
You don't need the song though as proven by the theatrical release. You can tell that Scrooge loved her and the scene leading up to the song is heartbreaking enough. Not to mention the song itself doesn't fit in with any of the other songs in the movie.