I get your drift, yet I doubt that you/your group would get much rest on such a trip: After driving through the wee hours, you'll arrive at your destination tired and sweaty, you won't have to access your hotel rooms (for a shower and changing clothes) before noon and on top of this you want to start a strenuous day of sightseeing just after sunrise? Don't you want to enjoy your trip?
Still, it's a good idea to avoid the peak hours, so maybe you should just consider shifting your plan a little. Motorway traffic starts ceasing once the commuters have returned home, i.e. roughly 6 p.m. (there will still be plenty of traffic, but less enough to start your trip). Either get an early dinner before that or plan a stop at one of the reststops along the motorways to get something to eat.
The landscape between Cologne and Berlin isn't very spectacular, so you won't miss that much at dark. Stop regularly to rest and/or change drivers (get a rental contract that allows at least one additional driver!). Consider stopping at Helmstedt/Marienborn, the former border crossing. With good luck and a flow, you might reach Berlin just before midnight. Book a hotel earlier and tell them in advance that a) you'll be arriving that late b) you'll need parking for a minibus. Be sure to know the hotel's precise location (again: navigation system would come handy, no fun getting lost in an unknown city at midnight...).
Finally: Get a good rest before starting the next morning.