It's not possible to date Karen after you've met Kuku, and attempting it will always get you a bad ending.
Karen does have a romance ending, but you can only access it before the stat raising part of the game starts.
As for raising Kuku's affection, just spam taking her to the cafe, then the date option when it unlocks, and you should be good to go!
Oh awesome thank you! I assumed there was no way to win the dialogue with Karen at the beginning cos it seemed like one of those obligatory game starting cutscenes that always goes the same way haha. That's a neat way to hide an ending!
And oh that's how you unlock dates! I was having trouble figuring out which places unlock which other places. I got the anime and the cat ones so now thanks to your help I have them all phew! Gonna try and play some more and hope I can get a better ending!