You are travelling between two Schengen countries, which means that you do not cross an international boundary.
You face two potential problems;
1 - the ID your airline requires to let you board the aircraft
2 - if stopped by the local police, you will not be able to provide proof of ID
Additionally, you will not be able to hire a car in Germany or do anything else that requires proof of ID - your UK driving licence is not proof of ID, as you have already been informed.
I agree with the last poster, get your passport couriered to you, travelling without it is not wise.