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The correct place to put the characters are Unicode Private Use Area (PUA) on positions U+E000 to U+F8FF (you have 6,400 code locations) on the Basic Multilingual Plane.
You also have PUAs on Plane 15 and Plane 16.
And, there are some icons here and there, like dignbats on positions U+2700 to U+27FF, and emoji, on positions U+1F30x to U+1F5Fx.
I’m considering doing something like this for my personal use.