+1
My idea: The original Ghostbusters should all be dead. Ray, Egon, Winston and Peter are all dead, right from the beginning of the movie. Not just Peter (because Bill Murray doesn't wanna do the movie) -- all of them. Sigourney Weaver's character is dead, too. They were all killed in some kind of horrible ghost incident/ghost fight that happened to them between Ghostbusters II and Ghostbusters III. Something major happened to all of them again and they didn't luck out and beat the ghosts, like they did in parts 1 and 2. They failed and they all died.
A new, younger cast of people come in and decide they want to open up the Ghostbusters business again. There's a lot of fear involved with people wanting to ghostbuster again because of what happened to the original Ghostbusters. But these new guys WANT to do it. They are fearless. They are brave. And they ain't scared of no ghosts.
Ray, Egon and Winston (and Peter, if Bill Murray agreed) pop in either in one scene or a couple of times -- as ghosts. Maybe just as supportive ghosts who are proud of the new people, or maybe they figure into the plot of the story somehow. But no matter what, they are dead, they are ghosts, they don't put on the Ghostbuster gear again and fight. Unless there's a cheesy ending and a ghost (one of them) destroys the other bad ghosts. Maybe they finally defeat the ghosts that killed them.
This way, Bill Murray (and Sigourney Weaver) don't have to appear in the film if Bill Murray doesn't want to because they could always say/joke that Peter and whatshername/Gate Keeper - DANA - they could say they're together in the afterlife and are perfectly fine with that. They gave up on the fight while the other dead Ghostbusters didn't, which is why they show up.