First, I should pass along that people who have more experience with perfect pitch than I do emphatically state that true perfect pitch (hearing a note and instantly knowing what pitch it is) cannot be learned as an adult. There may be agreement or disagreement, but that is what I'm told, and being in the field of music for years, I've never met anyone who's learned it as an adult.
Second, perfect pitch is not really necessary, and the only real benefit I can see between having absolute pitch and having 100% relative pitch is that perfect pitch makes a great party trick. It is just as useful to learn relative pitch (knowing all intervals and knowing all scale degrees by ear), and that can be learned. It sounds like the 7-note scale "hack" you're recommending is similar to learning scale degrees, which is extremely useful and doable.
Lastly, to qualify my statement, what you could learn is pitch memory. This is best learned by learning an intro/beginning of a song for each key and drilling yourself on it until you can hear it in that key in your head. Then when asked to sing a C, for instance, you play the beginning of your "C" song in your head, and you can hear and reproduce it. This is not true perfect pitch, and in my experience requires more work than it is useful. Also if you stop drilling the songs, your pitch memory will start to go -- it's not a one-and-done thing.
Bottom line, train yourself on relative pitch. I have completely accurate relative pitch, and I've never run across something that I absolutely needed perfect pitch to do.
Hope this helps!