It wouldn't likely be significantly different in a small lake. pH is spread almost instantaneously through a body of water. When titrating acid base solutions a single drop will cause the entire beaker to instantly change. pH or the hydronium ion isn't like something dissolved in the water. It can vary in the water like you can electrically charge some area of most matter like when you create static. Still there are variations in large bodies of water like the ocean where the pH is strongly influenced by photosynthesis and decay. Decay at the bottom tends to make it acid while photosynthesis tends to make it alkaline. There should be some differences in those locations if the lake if very large at all. The bottom is liable to relatively acid since acid is produced by decay and the upper portions of the lake where there is much photosynthesis are likely to be relatively alkaline.
