Well, I fixed the problem myself. In my previous example, I had:

$a !~ /\p{IsPrint}/
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which tests whether $a does NOT match any printing characters. (In other words, the test fails if it DOES match a PRINTABLE character.) So, what I wanted to do is close, but not quite the same thing:

$a =~ /\P{IsPrint}/
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which basically tests "does $a match the compliment of a printing character?" In other words, "are there ANY non-printing characters here?"

One has to look really closely and speak out the logic into a dark empty room before knowing intuitively which was the right choice ahead of time.