I tried something along the lines of #1 and #2 and it still causes an Apache2 child segmentation fault after a few successful requests. Here is the code I used:

sub RunCommand
{
    my $pgm = shift;
    my @execargs = @_;

    $SIG{CHLD} = 'IGNORE';
    # This should flush stdout.
    my $ofh = select(STDOUT);$| = 1;select $ofh;

    warn "Before first fork $$";
    my $kpid = fork;
    defined($kpid) || die;
    if ($kpid)
    {
        # Parent process
        my $w = waitpid($kpid, 0);
        # warn "waitpid returned $w\n";
    }
    else
    {
        close STDIN;
        close STDOUT;
        close STDERR;
        setsid() >= 0 or die;
        my $gpid = fork;
        defined($gpid) or die;
        if ($gpid) 
        {
          my $w = waitpid($gpid, 0);
          # warn "waitpid returned $w\n";
        }
        else 
        {
        open(STDIN, "</dev/null") or die;
        open(STDOUT, ">/dev/null") or die;
        open(STDERR, ">/dev/null") or die;
        # Child process
        exec($pgm, @execargs) or die;
        }
        CORE::exit(0);
    }
}
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I invoked that function from within my mod_perl2 response handler as:

&RunCommand("/home/dbooth/rdf-pipeline/trunk/pid.perl");
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The pid.perl command I ran uses no stdin, stdout or stderr, but merely logs its PID and date to /tmp/pid.txt:

#! /usr/bin/perl -w

# Append the PID and date/time to /tmp/pid.txt
# so that we can verify that this ran.

use HTTP::Date;
my $tmp = "/tmp/pid.txt";
open(my $fh, ">>$tmp") or die;
my $t = time2str(time);
print $fh "pid: $$ $t\n";
close $fh or die;
exit 0;
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P.S. See also the update to my original post, in which I note that this command still runs successfully during the HTTP request in which the Apache2 child process dies from a segmentation fault.