Rolf, I blame you for the sleep I lost over this!*

qr{
    (?(DEFINE) (?<loopfwd>
        (?{ print "pre:  $-[0]<$&>$+[0] pos:".pos."\n" })
        
        (?(<cur>) # we have a current character to inspect
            (?{ print "*CUR: <$+{cur}> / $-[0]<$&>$+[0] pos:".pos."\n"
+ })
            (?(?! .* \g{cur} )
                (?{ print " cur($+{cur}) doesn't repeat ahead, looking
+ back\n" })
                (?(?!(?&loopback))
                    (?<= (?<single> . ) )
                    (?{ print " cur($+{cur}) doesn't repeat behind eit
+her, "
                        ."FOUND SINGLE: '$+{single}'!\n" })
                    (*FAIL)
                |
                    (?{ print " cur($+{cur}) repeats behind\n" })
                )
            |
                (?{ print " cur($+{cur}) cur repeats ahead\n" })
            )
        )
        
        (?(?=.) # there are more characters, continue looping
            (?= (?<cur>.) (?&loopfwd) )
        )
        
        (?{ print "post: $-[0]<$&>$+[0] pos:".pos."\n" })
    ) )
    (?(DEFINE) (?<loopback>
        (?<=  # this lookbehind is fixed width!
            (?<prev>  (?! \g{cur} ) . )
            (?{ print "  prev: <$+{prev}> / $-[0]<$&>$+[0] pos:".pos."
+\n" })
            (?=(?&loopback)) .
        |
            (?= \g{cur} ) . .
        )
    ) )
    \A (?(?=(?&loopfwd))
        (?{ print "loopfwd matched (no single found)\n" })
        (*FAIL)
    |
        (?{ print "loopfwd didn't match (single found)\n" })
    )
}msx
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It turns out the trick is to anchor the string at the beginning and use zero-width assertions everywhere in order to implement a "loop over every character in this string" algorithm, and you implement the loops as recursion. And you can "loop" backwards from the current character with a "fixed-width" lookbehind!

The one thing I haven't figured out yet is getting the single value out of the regex. Another thing to note is that I had to invert the condition twice, since (*FAIL) breaks out of recursion but (*ACCEPT) does not. But I'll leave that for others, I'm done for now :-P

* Just kidding ;-) I slept quite well, which probably helped :-)