Sign and date the log book for before the end of the month. Do it already, rather than back date it infront of him. If he asks why that date is there, just say you did it when you knew you sold the car.
The DVLA do not have to receive the log book yet, so don't send it until you have the money.
They won't send out any fines right away and even if they did, you'd have a chance to inform the 'fine department' or whoever, of the change of ownership, pending the log book going through their system.