A pendulum clock ticks slower on top of a mountain and ticks faster at ground level, but an atomic clock ticks faster on top of a mountain and ticks slower at ground level.
Gravity affects pendulum clocks in the exact opposite way to atomic clocks.
Which clock is giving us the correct time? Are they both wrong?
Both give the correct time.
The frequency of a pendulum is:
$$ \nu = \frac{1}{2\pi}\sqrt{\frac{g}{l}} $$
Whatever your height above the Earth, if you measure $g$ and measure $l$ then use the calculated frequency as a measure of time, the pendulum will correctly measure time and its measurements will agree with the atomic clock. If you fail to take account of the fact that $g$ changes with height then you will indeed have an error in your time measurements, but that's due to your error and no fault of the clock.
The time you measure will indeed differ from the time measured at a different height due to gravitational time dilation, but this will affect both clocks equally.
