Hmm- greenish, turning to brown over time, clearing after aeration- is there any iron present in your water? If so, the greenish (iron II) material may be oxidizing slowly, which will lower the pH as a result. That would explain a slow pH reduction on sitting, and the clearing after aeration.
The difference between your "pure water" base titration result and your actual base requirement is the result of buffering capacity of the water. To understand what is buffering the pH (i.e. resisting the neutralization), we'd need to know more about your water composition, and have a titration curve.
Assuming there's no oxidation, re-dissolution or hydrolysis going on, straight acid/base reactions are relatively fast. The weak acid neutralization that's resulting in the buffering you're observing is somewhat slower, so maybe you need to move your pH probe further down the pipe. Mixing helps, and you should definitely put at least a static mixer in the line between your pH probe and your base addition point, preferably as close to the addition point as possible. This will slow the response of the pH controller, so you'll need to re-tune the controller.
If your flow is changing and you're measuring the flow, you should consider using a very simple model-based controller to de-couple the flow component from the pH control loop: i.e. if 50 gpm of flow requires 1 gal/hr of caustic, it stands to reason that 25 gpm of flow will require 0.5 gal/hr of caustic- you don't need to (and shouldn't) rely on PID parameters to decouple such a simple relationship. The PID parameters will then be used to control the actual variability of the water's composition, measurement variability etc.
In my experience, the number one problem with pH control systems isn't reaction rate- it's pH probe failure. From my experience, pH probes become slower and slower to respond the longer they're in service: i.e. they'll still read pH 4 in a pH 4 buffer, but you'll have to wait an hour or a day instead of a minute before the meter will show the correct reading. Inexperienced technicians don't notice, because they're impatient, and the probe will still seem to calibrate during a two-point calibration if it's particularly slow to respond, especially if they don't bother to check a third point. If they don't wait long enough for the reading to truly stabilize, they may push the "calibrate" button too soon- and not detect a defective probe. Depending on the design, the composition of the water they're in etc., pH probes can last periods of time from hours to years before they become too slow to respond properly and hence need to be replaced.