You can track the position of the black point then apply that data to a null. You can then use an expression to tie the position of the black point in the color stabilizer to the position of the null by simply dragging the pickwhip to the null's position property.
Repeat for the other point and you're done. Use a different point track for each sample point and make sure that the tracks are as accurate as possible by sampling a large enough area of the footage. AE's point tracker actually samples a group of pixels called a feature center inside a search area and not a single point. If the black point in your image does not have enough detail then you can pick something that has a lot of detail that is close to the black point and then offset the attach point before you start tracking. Adjustments to perfect the track can be made by moving the Attach Point after the track is complete.
Yes! Ok, great advice.
I tracked the black point using the method you suggested in the above picture. But I am not familiar with expressions and what you meant by "repeat for the other point."
Using that thinking though, I copied the track points of the Null object's Position and pasted them to the Color Stabilizer's black point Position, and it worked! So not exactly as you suggested but your help was instrumental. Thanks a lot! The issue is better but not completely solved but that's the effectiveness of Color Stabilizer, not your excellent suggestion.