Update: Install a kernel with large file support enabled.

Untested code:

open my $infile, '<', '/path/to/basename.ext' or die $!;

# count the >'s and get average number for each split file
my $total = 0;
$total += tr/>// for <$infile>;
$each = $total / 8;

# go bach to start and write first seven split files
seek $infile, 0, 0 or die $!;
for (1..7) {
    my $splitcount = 0;
    open my $outfile, '>', "/path/to/basename.split${_}.ext" 
        or die $!;
    $splitcount += tr/>//,
        ($splitcount >= $each ? print $outfile $_ : last) 
            for <$infile>;
    close $outfile or die $!;
}

# put the rest in split #8
open my $outfile, '>', "/path/to/basename.split8.ext" 
        or die $!;
print $outfile $_ for <$infile>;
close $outfile or die $!;
close $infile or die $!;
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This will leave the first seven with slightly more than the desired average, and the last several short.

After Compline,
Zaxo