Do you want to close Firefox with closing the last tab?

If you set browser.tabs.closeWindowWithLastTab to false then you get a close button and can close this tab, but Firefox doesn't close of course with this setting.

I would like to keep the current functionality of closing the windows when closing the last tab via keyboard shortcut, and have it apply to closing tabs with the middle mouse button.

As it stands, middle clicking the last tab has no effect on the tab or window. other tabs close as normal.

You can only close the last tab by setting that pref to false.
You can't have a different behavior for the mouse and the keyboard (there is a close Window shortcut key).
Otherwise you need open a new tab and drag that tab to position one to have the first tab as an empty tab.
If you really want to close the tab then best is to first close the tab and then close the window to prevent a History > Recently Closed Windows entry.

See also Help > Keyboard Shortcuts