You should be fine with your current approach. While German law requires you to have an ID, the law does NOT require you to have the ID with you at all times. With your passport and your paper copy you have enough data about you that in normal cases nobody would insist on seeing the original (Formally, if you don't have the original police would have the right to accompany you back to your hotel to check the ID).

In general, I've had to show my ID much less often in normal life (=when I'm not traveling by air) in Germany than in the US anyway, I can only remember one instance in the past 25 years where I was stopped and asked for my ID. This was part of a very large search for an escaped criminal, and I think even there the driver's license would probably have been sufficient.