Does this match the style you're aiming for?
https://i.sstatic.net/q7eYQ.png
If it does, you can organize the main button and dropdown section of the button within a button group like so:
<div class="btn-group">
<button class="btn btn-default" onclick="window.location.href='#'">
Split Dropdown
</button>
<button data-toggle="dropdown" class="btn btn-default dropdown-toggle">
<span class="caret"></span>
</button>
<ul class="dropdown-menu">
<li><a href="#">Dropdown Menu item 1</a></li>
<li><a href="#">Dropdown Menu item 2</a></li>
</ul>
</div>
Here is a demonstration of it in action. (Remember to include the Bootstrap and JQuery resource links in your own project.)
UPDATE: Here is a similar approach integrated into panels.
This is how it appears:
https://i.sstatic.net/wwKJO.png
And here is the relevant code (full code on fiddle):
<div class="container">
<div class="panel with-nav-tabs panel-default">
<div class="panel-heading">
<ul class="nav nav-tabs">
<!-- Other tabs here -->
<li class="dropdown btn-group">
<a class="btn" href="#tab-main" data-toggle="tab">Tab w/ Dropdown - Main Tab</a>
<a data-toggle="dropdown" class="btn dropdown-toggle">
<span class="caret"></span>
</a>
<ul class="dropdown-menu" role="menu">
<li><a href="#tab-dropdown1" data-toggle="tab">Tab w/ Dropdown - Dropdown 1</a></li>
<li><a href="#tab-dropdown2" data-toggle="tab">Tab w/ Dropdown - Dropdown 2</a></li>
</ul>
</li>
</ul>
</div>
<div class="panel-body">
<div class="tab-content">
<!-- Other panes here -->
<div class="tab-pane fade" id="tab-main">Tab w/ Dropdown - Main Tab</div>
<div class="tab-pane fade" id="tab-dropdown1">Tab w/ Dropdown - Dropdown 1</div>
<div class="tab-pane fade" id="tab-dropdown2">Tab w/ Dropdown - Dropdown 2</div>
</div>
</div>
</div>
</div>
The potential issue I notice with this method is that if you are on a dropdown panel, you cannot return to that tab's main panel without selecting a different tab first. This happens because both the main and dropdown tab receive the active
class, which deactivates the tabs (aside from applying styling that shows the tab is chosen). If this is problematic, you may need to use some JavaScript to solve it.