Kopech's a bit easy to fluster when he's on the mound, so I'm sure him giving up that many homers on what he thought was some good pitches was messing with his head. It's true that only one curve was actually hit for a homer, but when you can easily eliminate one pitch based on a tip, it helps a lot. He was also only throwing the change to lefties and I don't think Conforto got a single one so it was pretty easy to tell when he was gonna get a fastball. And like you said, he was hanging the slider a lot, so non-flared glove was either a heater or a bad slider. And a flared one was a curve.
What's worse is that nobody on the staff apparently fixed the issue. They maybe just told him to stop throwing the curve since he only threw it once yesterday. But the tip is still obviously there.
Curve:
Non-Curve: