DUH! Okay, I should have seen this before, but there's a Set() function:

$thumbnail->Set(depth => 8);
[download]

So, to ensure that my code performs as expected on both systems:

#!/usr/bin/perl

use Image::Magick;

my $thumbnail = Image::Magick->new;
$thumbnail->Read("full.bmp");
$thumbnail->Set(depth => 8);
$thumbnail->Write("full.png");

$thumbnail->Resize(width=>'200', height=>'150');
$thumbnail->Write("tn_200x150.png");

$thumbnail->Resize(width=>'100', height=>'75');
$thumbnail->Write("tn_100x75.png");

undef $thumbnail;

exec "file /home/mhenkel/*.png";
[download]