Jake,
I’m not sure about what you want to do at the end of the sequence. I found several DPDT push on/push off switches from All Electronics for less than a dollar a piece. Wire them in sequence with a NO of the current switch and NC of the next in the sequence and it will do as you asked. Since these are DPDT, you can use the second pole to extend the sequence for a second cycle. At the end of the first cycle, all switches are down and lamp 5 is lit, at the end of the second cycle, all switches are up and lamp 5 is lit.
However, if this is to be a presentation display, it is easy enough to have all lamps off if all buttons are up and only lamp 5 on if all buttons are down and you have to reset by pushing the down buttons a second time.