Why didn't they work together to the degree that Austin and McMahon did? Because they legitimately hated one another and WCW was obsessed with killing Flair's heat once Hogan came into the company.
Bischoff sued Flair for breach of contract because he missed a Nitro to go see his son win a wrestling tournament, despite Flair having given them advance notice that he was missing the show for that exact reason. It's why Flair was off of TV for half of 1998. And besides, they DID actually try to capitalize on it: they had a match at Starrcade 1998 that built off their real life issues (Bischoff won after Curt Hennig interference, then Flair won the rematch on the following Nitro to become President of WCW). It just wasn't actually that good and was overshadowed by the Fingerpoke of Doom, so nobody really remembers it.
By the way, this entire situation is the reason why this promo happened (skip to 10:30 or so):