To see the lever, press a number key that you do not have hot-keyed to a group. For example, I hotkey "0" to my space station when I'm defending, but since I'm there attacking if I have the Death Star with me, then "0" is left unassigned.
When the countdown hits zero, I press zero. All of my other ships are deselected and the only thing on my unit bar is the Death Star lever.
And after firing I never wait for the cinematic because the battle is still going. When the Death Star cinematic starts, press the space bar or double-click to get back to the battle, then select and manage your groups as usual.