>perl -wMstrict -le
"my $s = 'Foo|Fate|||Fate||||Pre|Fate|Post|||||Alive|Alive';
 my $Fate = qr{ Fate }xms;
 my $not_Fate = qr{ (?! $Fate) . }xms;
 $s =~ s{ (?: $Fate $not_Fate*){2} \K $Fate }{}xms;
 print $s;
"
Foo|Fate|||Fate||||Pre||Post|||||Alive|Alive

##</code><code>##

>perl -wMstrict -le
"my $s = 'Foo|1ate|||2ate||||Pre|3ate|Post|||||4ate|Alive';
 my $Fate = qr{ \d ate }xms;
 my $n = shift;
 my $x = 2 * $n;
 $s =
   join '',
   grep defined,
   (split /($Fate)/, $s, $n + 1)[0 .. $x - 2, $x]
   ;
 print qq{'$s'};
" 3
'Foo|1ate|||2ate||||Pre||Post|||||4ate|Alive'