How to go about making a Core Tab on the Contact Summary screen view Inline instead?
Rather than smarty, JavaScript will do the trick.  Those tabs are all separate pages, and you can load any CiviCRM page/form as a snippet instead.

First, in your extension, use a hook to include the Javascript and provide the contact ID for it:

$r = CRM_Core_Resources::singleton()

// Provide the contact ID (assuming you've got it handy)
$r-&gt;addVars('memberTabInline', array('contactId' =&gt; $contactId));

// Add the script
$r-&gt;addScriptFile('com.example.membertabinline', 'membertabinline.js')


Then in membertabinline.js:

CRM.$(function($) {
  $.get(CRM.url('civicrm/contact/view/membership', {
    reset: 1,
    force: 1,
    cid: CRM.vars.memberTabInline.contactId,
    snippet: 1
  }), function(result) {
    $('#mainTabContainer').append(result);
  });
});


You'll probably need to add a div or something and have this fill it so that you don't have the membership stuff appending multiple times, but this is the gist.

Alternatively, you might look at https://wiki.civicrm.org/confluence/display/CRMDOC/Ajax+Pages+and+Forms to see if you can fill a wrapper with the tab and have it work.  I'm only familiar with this re: popups, but it looks like it might work within any element.