To be specific, in your view the innermost column headers are at the bottom, they are just not shown (the Entity discrete field). You can Ctrl+drag a copy of Measure Names onto the Columns Shelf after Entity, but that will result in showing the Measure Names headers for each and every bar, whereas I'm guessing you only want one instance of each measure name header per pair of bars.
I think your best bet is to use a dashboard with a vertical layout container, with one worksheet that shows the data and has the Measure Names hidden at the top, and a second worksheet just below it that has the Measure Names header.
Thank you for your response. You are right when you say I don't want the measure names to appear twice at the bottom. And yes, I hid the headers for the entity column which were at the bottom. I saw that same solution in another forum to add a text box in the dashboard, but this is pretty tedious to make sure that the text lines up with each column. You should be able to drag the top headers in the worksheet to the bottom of the bar chart if you hide the inner most column headers.
I agree, aligning pixels does get tedious at times. If you create an idea in the Ideas section, I'd vote for it! A couple questions that would be worth answering, though, are what should be done when another discrete value is added to the view, and what to do when a discrete value is removed from the view? In other words, what header location choices should Tableau respect and what should Tableau change as pills are dragged and dropped from the view?