Should I carry my passport for a domestic flight in Germany

There is an FAQ on the federal ministery of the Interior about residence permits. Unfortunately, one can not simply switch to english, and I didn't find it on the english site.

It states:

Muss ich immer meinen Pass oder Passersatz mit mir führen?

Nein, aber Sie müssen auf Verlangen den Ausländer- und Polizeibehörden Ihren Pass bzw. Passersatz und Ihren Aufenthaltstitel vorlegen.

[...]

• Wenn Sie keinen Pass besitzen, diesen auch nicht zumutbar erhalten können oder Ihren Pass vorübergehend einer deutschen Behörde überlassen haben, müssen Sie umgehend einen Ausweisersatz beantragen.

which translates to

Do I always have to have my passport or replacement with me?

No, but upon request of police or aliens authorites, you have to present your passport or replacement and your residence permit

[...]

• If you do not have an ID, you can not get it with reasonable effort, or have passed it temporary to a German authority, you must apply for a replacement passport.

So, it boils down to: You don't need to have your passport with you, but if police asks, you have to present it...

ID card and passport are the only official documents to confirm your identity, the residence permit is only a supplement.

There is no general passport control for flights within the Schengen area and you can often board an aircraft without showing your passport anywhere. Airline staff may be fine with anything looking official, like residence permit or drivers license. But it is still possible that police asks you for your passport. Maybe, you get away with your story, maybe, not.

(Within this context, it is interesting how a consulate can take your passport, leaving you without any official ID. As German, you can easily get a replacement as requested by the last paragraph of the quote, but for others... hmmm)

Sorry, I don't have a solution for your particular problem, this is just the law.

On intra-Schengen flights, there is no systematic, police-performed checking of identity documents which would require a passport or ID card or similar. And moreso for truly domestic flights (crossing a Schengen border you are de jure required to have ID with you even if it isn’t checked. Within Germany you are typically not required to have an ID on you all the time).

However, either the airline or the airport may want to confirm that you are indeed the traveller that should be travelling on the ticket in question. For that, they may want to see some form of identification.

Unfortunately, I couldn’t find anything definite on Lufthansa’s website as to which documents are accepted for domestic flights (they only go into detail for international flights). It is entirely possible that given your circumstances your residence permit (which is government-issued) or a driving license (with a picture) would suffice.

The definite answer can only be given by the airline, though.

My girlfriend is a foreigner with german residence card, and while I advise her to take her passport on all our trips, she almost never needs to show it unless we cross a non-Schengen border.

On flights within the EU, passports or ID cards are rarely checked at all. Even less so on national flights. At HAM, which I've used regularily, ID is rarely requested for national flights, but I've seen it happen (not to individuals, rather a "please show your passport and boarding pass when boarding" announcement).

So practically, you should be fine. That doesn't mean the airline can not cause you trouble, either legally or illegally. Many people have been denied entry into Schengen countries, sued and won. So to be certain, you should ask the airline, in writing, and get a written response that you carry with you.