So I need people to hear me out for a second.
I'm a fan of Jason actually being Lady Shiva's kid for a couple of reasons. Mostly because I think it would set up some really fun and interesting dynamics between Cass, Jason and well Shiva and Talia. But generally it's a guilty pleasure of mine as I can't typically justify in my head how Jason ended up in Gotham with Willis (Or how Willis would even know Shiva). But this is where you need to hear me out there is one universe where i think it would work though and that's Batman the Adventures Continue the spin off comic of Batman the animated Series.
Jason's story is already so different in that because well... Tim kind is Jason in BTAS. Any Jason is introduced to us by fighting off a gang by himself. And we aren't given a lot on Jason's background other then he was an orphan and had a brother who died because of the gang he decided to beat up. So that ambiguity does give more room for things to be reworked and Jason being more naturally good at fighting would lend a little more creditability to being Shiva's son here.
Now you might counter point me with Jason having a brother. One he clearly cared about because he decided to take on a whole street gang for him and eventually named himself after (The Red Hood) but we could just give ourselves another little Catherine situation where Jason just ended up with his brother and/or his brother's family before they passed.
To my memory Shive doesn't show up in BTAS or any of the DC Animated Universe. But think about it. Red Son Rising ends ends with Jason telling Bruce to be with the Robin who needs him and getting carried away by a current only to end up with Slade. There is already potential there with Slade pretty clearly wanting take over Batman's mentor role and potentially even get to Batman through Jason based on what happened in Mentors.
And I hope your still with me here because here is how I'm thinking it would go. Slade starts noticing things about Jason's skill and natural ability. And Having already heard about Shiva having a kid starts looking into it. Tell Jason all this trying to play on Jason's desire for a family who would accept him as is and trying to earn his trust through this. You could also at the same time have Cass appearing in Gotham. And having Bruce being reminded of Jason in Cass's natural gifts. You could have Bruce running her DNA and having a sibling match to Jason here and you could work in an angle of Cass feeling the need to bring Jason back not for just herself but to fully be accepted by Bruce. You could just do so much with the conflicts between Slade and Batman, then Jason and Batman and Slade and Jason. Because I think Jason wouldn't be that trusting of Slade here but Jason remembering his brother and learning of more family I think he'd go for it regardless, then is conflicted with Shiva and Cass being with Bruce.
So lot of good character conflict. Lots of new dynamics and to me a pretty reasonable set up for it. And Jason's dad can be who ever! Some random thug, Willis or Joe Todd, David Cain, Richard Dragon pick your poison. I just need more Cass and Jason (And letting me ignore how they went THE ANGRY ROBIN route again.)