OP here. Yeah, "copyright law" was a lazy shorthand, but it reads better than "tortious interference."
PureGym's T&Cs [1] have a ridiculously long "PIN abuse policy" (probably meant to stop people sharing with mates). They can cancel memberships or even retroactively charge for gym use if you "knowingly provided your PIN to another individual."
I'm not a lawyer and don't fancy being the test case for whether entering your PIN on a third-party website/app counts as "knowingly providing" it. Given how their app works, I suspect they might just ban a bunch of accounts instead.
Though now that I think about it, the squat racks are always packed, so maybe I should just distribute the app to people who go at the same time as me.
[1] https://www.puregym.com/membership-terms-conditions/