I don't think you can get a LAPL medical yet?
No, not quite. You will have to wait until Sept 17 . I spoke to my AME today to book one, but he doesn't yet know exactly what it involves so wouldn't give me an appointment until early October so he has a chance to get up to speed with it. Forget the GP route- there is absolutely no incentive for them to register to do something they don't understand and are not interested in.