How can I move a module block's title to display within the block's template?
You can do this by overriding the block template:

create a template called block--block-name.tpl.php OR override all block templates by using block.tpl.php and move the code the prints the title into the content div:

&lt;div id="&lt;?php print $block_html_id; ?&gt;" class="&lt;?php print $classes; ?&gt;"&lt;?php print $attributes; ?&gt;&gt;

  &lt;div class="content"&lt;?php print $content_attributes; ?&gt;&gt;
    &lt;?php print render($title_prefix); ?&gt;
      &lt;?php if ($block-&gt;subject): ?&gt;
       &lt;h2&lt;?php print $title_attributes; ?&gt;&gt;&lt;?php print $block-&gt;subject ?&gt;&lt;/h2&gt;
      &lt;?php endif;?&gt;
    &lt;?php print render($title_suffix); ?&gt;
    &lt;?php print $content ?&gt;
  &lt;/div&gt;
&lt;/div&gt;