Display Node edit under edit tab
I suggest hide default edit tab and create a tab set with Quick Tabs module.I mean bypass default drupal menu tab( or hide it completely ) and create your own . this module help you to create a quicktab set that click on each item don't need page getting refresh.


  The Quick Tabs module allows you to create blocks of tabbed content,
  specifically views, blocks, nodes* and other quicktabs*. You can
  create a block on your site containing multiple tabs with
  corresponding content. Clicking on the tabs makes the corresponding
  content display instantly, using jQuery. With the 6.x-2.x branch, this
  can be done in either ajax or non-ajax mode:

if you choose ajax mode for your Quick Tabs block (and this is chosen on an individual QT basis), only the content of the active tab

  
  will be loaded on page load and the content of the other tabs will be
  loaded via ajax on tab click (this is good for performance if you are
  loading, for example, several large views)
      in non-ajax mode the tab content is just hidden / shown on tab-click. This latter method is an ideal way to do something like the
  Most Popular / Most Emailed stories tabs you see on many news
  websites, where it is used more as a space-saving device in your
  sidebar. If however, you want a QT block as the main content of your
  entire page, then the chances are that ajax mode will be more
  appropriate.
  
  Once created, the Quick Tabs blocks show up in your block listing,
  ready to be configured and enabled like other blocks. Multiple Quick
  Tabs blocks can be placed on a single page.
  
  For theming of the tabs, several styles have been provided for you to
  choose from (in the 7.x-3.x branch this functionality comes in a
  separate submodule, Quicktabs Tabstyles), along with the default
  option of no style, if you prefer to add your own theming instead.