(Note to self: V 10.1 workbook)
See attached. Forget about swapping dimension names. Swap sheets instead.
In the attached I made a dashboard. Dashboard 1 has your two original sheets floating on it.
For Dashboard 2 I made copies of your two sheets and added a filter to display (or not) the sheets depending on which parameter value is selected. See Dashboard 2. On that dashboard I added a container and dragged the two sheets into it. Play with the "View By" parameter and see how the two sheets interact with it.
Go to Dashboard 3. Here I removed the titles of the two sheets. Now, when a sheet is blank, it collapses and the remaining sheet shifts to the left side of the container.
Dashboard 4 just formats the container a little bigger, and I added a text box that displays the value of the selected parameter.
The technique here is called sheet swapping.
This is a tangential question to your solution, but can this be achieved with fixed rather than floating sheets?
The key to sheet swapping is putting the two sheets into a container. And my personal preference is to use floating stuff all the time. However, you could put the two sheets into a container, fix the size, and then drag that container into your tiled dashboard and the sheet swapping should still operate the same way.