Your OP node title specifically refers to combining regexes, so here's an approach that decomposes what seem to be the essential elements of your regex and re-combines them to form the final matching regex. I find a decompositional approach makes it easier to think about a regex (especially a complex one) when writing it, and to maintain it later. (Note: Some of your StackOverflow examples have leading characters before the  $mir pattern. If this is really the case, eliminate the  \A absolute-beginning-of-string anchor from the matching regex.)

Another Note: If it's just a matter of excluding anything matching  <EXP-N-\d+> then BrowserUk's 'simpler' solution here is by far the best.

>perl -wMstrict -le
"my @strs = qw(
   <MIR-1><EXP-V-3><VACCVIRUS-PROP-1>
   <MIR-1><ASSC-PHRASE-1><VACCVIRUS-PROP-1><PATTERN-1>
   <MIR-1><EXP-V-0><ART-0><VACCVIRUS-PROP-1>
   <MIR-1><EXP-V-0><ART-0><BE-V><VACCVIRUS-PROP-1>
   <MIR-1><EXP-V-0><EXP-N-0><VACCVIRUS-PROP-1>
   );
 ;;
 my $tail = qr{ \d+ > }xms;
 ;;
 my $mir      = qr{ < MIR-            $tail }xms;
 my $exp_v    = qr{ < EXP-V-          $tail }xms;
 my $exp_n    = qr{ < EXP-N-          $tail }xms;
 my $assc_phr = qr{ < ASSC-PHRASE-    $tail }xms;
 my $vaccvir  = qr{ < VACCVIRUS-PROP- $tail }xms;
 ;;
 for my $str (@strs) {
   print qq{'$str'} if $str =~
     m{ \A
        $mir
        (?: $exp_v (?! $exp_n) | $assc_phr )
        .*?
        $vaccvir
      }xms;
   }
"
'<MIR-1><EXP-V-3><VACCVIRUS-PROP-1>'
'<MIR-1><ASSC-PHRASE-1><VACCVIRUS-PROP-1><PATTERN-1>'
'<MIR-1><EXP-V-0><ART-0><VACCVIRUS-PROP-1>'
'<MIR-1><EXP-V-0><ART-0><BE-V><VACCVIRUS-PROP-1>'
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