I never got the impression that Oscar was their love child...unless Dana did a Carol Marcus and kept Oscar away from Peter and lied on top of it, I think Peter would be pissed about that especially the way Dana and his relationship went and all the grief that Dana had given him about his antics during their relationship. I realize that we pretty much only got a throwaway line about their relationship. Now if they wanted to have Oscar as the lead is fine and introduce a character that is actually Peter and Dana's son explaining that they got married at the end of Ghostbusters II and had a son together that's fine as well. This movie has been in development hell for so long now I'm starting to doubt it will ever happen.
I also never got the impression that Oscar was Venkman's son. As I recall, they stated that Dana had gotten married after she left Peter and that she and her husband had recently broken up. And that Peter and Dana had not seen each other since their break up. So clearly Oscar was the child of Dana's off-screen ex-husband.
But I could see that Oscar winds up being Venkman's step-son. And that Oscar considers Peter to be as much his father as his birth daddy. Peter could even have adopted Oscar, giving him the Venkman surname.