At German airports normally there are no document checks. AFAIR neither in Sweden. If there is, it's the airline and a driver's licence may suffice (I don't know much about the residence permit - if yours look like this, it might do the job, too)
I would do the trip without thoughts, but I'm "at home" within the EU, so my threshold can be lower than foreigners'
Why not make also a photocopy of your passports data page?