Starlink satellites are not dumb signal repeaters. They also use relatively localised spot beams. But yes, anything you transmit through one probably can't be localised to within more than about a hundred km, if the only available information is which beam you're in at which time.

You'd need to avoid providing information about the time the beam crosses over your position, which means you'd only activate your connection sporadically, at carefully planned times. You might pick a location relatively near you and down-orbit from you, and connect when that location comes into view of a new spot beam, and disconnect when it's directly over that location, perhaps.