index does a check for literal characters. By specifying you want CAT but not CATALOG, you are adding criteria that cannot be expressed using literal characters only. You could get closer by using $KEY = " CAT ", since your string is whitespace delimited, but this would fail if CAT is either your first or last entry.

The only way to implement this robustly would be to use a tool that can check for non-literal character criteria, e.g. regular expressions. For example, you could use the word anchor \b to specify you want word boundaries on either side of CAT -- see Using character classes. You can then use $-[0] (see @-) to learn the position of that match. So you might say

my $string = "CATALOG SCATTER CAT CATHARSIS";
my $key = 'CAT';
$string =~ /\b\Q$key\E\b/;
print "$key found at $-[0]\n";
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