The only way you can arrive at the conclusion that one clock should be 24 hours into the future is if you assume that time is static for all observers -- i.e. one 24 hour period for the observer at the peak is exactly another 24 hour period for the other observer at the base. Stated another way, you are exchanging their hours 1 for 1, i.e. a 24 hour period for the peak observer is 24 hours for the summit observer. Following this line of thought, one could rationally conclude that the peak observer should be 24 hours in time ahead of the base observer. And you did just that. But this is a fatal misunderstanding of what relativity teaches, and so naturally you arrive at the wrong conclusion.
Relative time means that in the same number of sunrises and sunsets, each observer actually experiences a different interval of time as measured by whatever clock you chose (dripping water, heart rate, atomic clocks, mechanical clocks, etc.) You seem to be confused on this point -- it doesn't matter whether its an atomic clock or not. It can be any process which changes as a function of time (that includes your existence and all of the interactions with the environment you have). What relativity means is that the length of the day (i.e. sunrise to sunrise) is actually slightly different for each observer.
Again, the summit clock's readout would suggest that there was an additional sunrise if and only if you assume the summit clock experiences the sun rise at the same precise interval the base clock does: once every 24 hours. Relativity makes no such assertion. In fact, it says quite the opposite: each observer will experience a different amount of time between each sunrise. So when calculating out how many sunrises should have been seen by a Summit Clock, simply take the time between sunrises and divide it by the time experienced by the observer. When this calculation is done, both observers will agree on the exact number of sunrises regardless of the time they experienced. What they won't agree on is how much time elapsed between each sunrise and how much total time was experienced. This is what it means to say that time is relative.
It also doesn't seem to account for the fact that rotational speed is different for each clock. The one at the summit would be moving faster through space to cross the vector at the same "time"