It turns out the height of the menubar window was manually being set to 30 using $toplevel->geometry. Having it use height 0 instead eliminates the empty space. I then have to tell the program separately to pretend the menubar window has a height of 30 so that the positions calculated for the other windows do not overlap with the menubar window. Since I'm using geometry anyways, I can omit the spacer widget. Here's what that looks like for the test program:

use warnings;
use strict;
use Tk 800.000;
my $mw = MainWindow->new;
my $menu = $mw->Menu();
my $file_menu = $menu->cascade(-label => 'File', -tearoff => 'false');
my $edit_menu = $menu->cascade(-label => 'Edit', -tearoff => 'false');
my $help_menu = $menu->cascade(-label => 'Help', -tearoff => 'false');
$mw->configure(-menu => $menu);
# same effect as geometry('200x0+0+0')
#my $spacer = $mw->Frame(-width => 200)->pack;
$mw->geometry('200x0+0+0');
# resize handle obscures Help menu
$mw->resizable(0,0);
MainLoop;
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