I think my interpretation is (after reading Hopes' response) it wouldn't make sense for Quinn to be working for DeLuca's rival families, because you gotta consider the way all characters are and the world they're in for each game.
Like Hopes said, in DeLuca, the whole world and basically all characters are exaggerated. Whether this exaggeration is in regards to one's strength, agility, intelligence, personality, (or any other distinct trait) etc depends on the character. A lot of times, it's hyper, amped up.
Compared to that, BaDIK is basically slice of life, its world focused very much on realism (except MC's conquests of girls/women
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).
With a crossover (not funsie, but crossovers of characters actually going from one game to another game, canonically), it happens with the assumption that the 2 games are in the same universe, and that's just not possible with these two games (also, my guess is that even though not MCs [or even side character for that matter], drug dealers/junkies in DeLuca universe would be much more dangerous than the drug dealer drug dealers/junkies in BaDIK world)
It just doesn't gel at all with each other due to how different they are (in fact, it's the same case with DeLuca and any other game).
Of course, I'm also not free at all from the fault of thinking the same way as you either (slapping my own forehead in shame), throwing around my "Serpent crossing over to another game" or "possible Alexandra & DeLuca crossover" silly/absurd ideas before Hopes explained well in why it wouldn't work.
And I'm fine with that (just keeping it as "What if" but non-canon scenarios in the head). Funsie crossovers as Hopes mentioned in his reply is all good.