You can still go to Lap 2/3 even when Pizza Time is over. Select Pizza Timer from the taskbar to focus the Window, and then press the numbers.
The time's up screen was going to appear when DreamTime was off and no custom command was entered. I ended up not implementing it because I couldn't get the text falling animation right, however, reading your comment just gave me the idea to add a toggle to show the "Time's Up" screen, and then do what they have to do, or to just do it correctly. I'll try to do the anim once more to see if I can do it.