I've only known a few lurchers in my time, and they have always chased cats (at some speed).
It's not so much the fluffy, but the flight that triggers their natural response.
I did know a lurcher-terrier cross that lived happily with a cat, but it was a cross and like your situation, the cat was their first so held dominance in the household.
If you're taking on from a puppy stage I'd like to think they'd become accustomed to each other and the cat wouldn't run away as they grew up together.