So I do (think I) need to pass through the file twice - once to find the references I want to remove, and once to actually remove them.

I took that as a challenge ;-) This only needs a single pass by reversing both the input and output by piping it through tac, and produces your desired output:

use warnings;
use strict;

die "Usage: $0 INFILE\n" unless @ARGV==1;
my $INFILE = shift @ARGV;

open my $ofh, '|-', 'tac' or die "tac (out): $!";
open my $ifh, '-|', 'tac', $INFILE or die "tac $INFILE: $!";
my ($aminblock,$prevnum,$foundstr);
my %found;
while (<$ifh>) {
    chomp;
    my $out=1;
    if (!$aminblock) {
        if (/^end$/) { undef $foundstr; $aminblock=1 }
        elsif (/^\s*(?:foo\s+)?ref\s+(\d+)\s*$/) {
            die "ref $1 without block?" unless exists $found{$1};
            $out = !$found{$1};
        }
        else { die "unexpected outside of a block: $_" }
    }
    else {
        if (/^\s*(\d+)\s*$/) { $prevnum=$1 }
        elsif (/^begin$/) {
            die "block ended without number?" unless defined $prevnum;
            $found{$prevnum} = $foundstr;
            undef $prevnum;
            $aminblock=0;
        }
        else {
            undef $prevnum;
            if (/bar/) { $foundstr=1 }
        }
    }
    print {$ofh} $_, "\n" if $out;
}
close $ifh or die "tac $INFILE: ".($!||"\$?=$?");
close $ofh or die "tac (out): ".($!||"\$?=$?");
[download]

Although the two passes through tac might actually make that less efficient for large files. Here's a two-pass version:

use warnings;
use strict;

die "Usage: $0 INFILE\n" unless @ARGV==1;
my $INFILE = shift @ARGV;

use constant { STATE_IDLE=>0, STATE_BEGIN=>1, STATE_INBLOCK=>2 };

open my $fh, '<', $INFILE or die "$INFILE: $!";
my %found;
my $state = STATE_IDLE;
my $curnum;
for my $pass (1..2) {
    while (<$fh>) {
        chomp;
        my $out = 1;
        if ($state==STATE_IDLE) {
            if (/^\s*(?:foo\s+)?ref\s+(\d+)\s*$/) { $out=!$found{$1} }
            elsif (/^begin$/) { $state=STATE_BEGIN }
            else { die "unexpected in state $state: $_" }
        }
        elsif ($state==STATE_BEGIN) {
            if (/^\s*(\d+)\s*$/) { $curnum=$1; $state=STATE_INBLOCK }
            else { die "unexpected in state $state: $_" }
        }
        elsif ($state==STATE_INBLOCK) {
            if (/^end$/) { $state=STATE_IDLE }
            elsif (/bar/) { $found{$curnum}=1 }
        }
        else { die "bad state $state" }
        print $_, "\n" if $pass==2 && $out;
    }
    die "unexpected state at eof: $state" unless $state==STATE_IDLE;
    seek $fh, 0, 0 or die "seek $INFILE: $!";
}
close $fh;
[download]

Update: Note that these solutions don't remove ref N lines if they appear inside begin...end blocks; this was an assumption I made, but it's actually unclear what the desired behavior is in that case?

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