A simple optimization would be to replace

for (0..($strLen-1))
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with

for ( 0..$strLen/2 )
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One could get fancier by testing only those cycle lengths that are either 1 or prime factors of $strLen, which could make a difference for long strings.

Also, I think it's simpler just to test for equality of $str and $copy, obviating the xor and the regex:

for ( 1..$strLen/2 )
  {
  $copy .= substr $copy, 0, 1, '';
  return wantarray  ? ($_, substr $str, 0, $_) : $_
    if $copy eq $str;
  }
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