They will both show acceleration. The one on the mountain will show an upward acceleration from the mountain pushing against the device against the pull of grav

They will both show acceleration. The one on the mountain will show an upward acceleration from the mountain pushing against the device against the pull of gravity. The rocket would too, but presumably greater if it’s accelerating faster than gravity is pulling. The key is that the gravity acceleration vector never shows up in any context. That’s because it’s not a force and the perceived acceleration is an effect of relativity, caused by the gradient of time and space. It’s entirely counter intuitive to our understanding of force and acceleration that you can observe increasing relative motion without a force based acceleration, but there it is