Oki Doki! The culprit is the output buffering. When executing the script via command line, the header does display exactly after 8192 bytes, which when run through Apache produces an error. Output buffering can be disabled in Perl, so I added the line "$|=1;" to my script and now it works.

#!/usr/bin/perl
use strict;
$|=1;
use HTML::Template::Pro;
my $tmpl = HTML::Template::Pro -> new(filename => 'test.tmpl');
print "Content-Type: text/html\n\n";
$tmpl -> output();