Max Martin also did the chorous and one verse to Shake It Off for Katy about shaking off the fact she has no Grammys and how people always come for her about it and that was a Prism reject but then he offered it to Taylor and she took it.
So what if Bad Blood was a demo? It's not like she took it and merely recorded it. She definitely re-wrote the already written parts and stuff like that.
So what if it was a Katy Perry demo? Producers pitch their work to their artists of course. Taylor choosing it must've been some calculated way of getting at Katy.