DUH! Okay, I should have seen this before, but there's a Set() function:
$thumbnail->Set(depth => 8);
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";
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