Cycle rental Europe

A one-way bike rental from Copenhagen to Berlin? Doubtful, but search away. There's a self-guided tour per your route at BikeTours.Com (AKA Bike Tours Direct), with a 5% discount if you are a member of Adventure Cycling Association.

A couple of suggestions from www.bicyclegermany.com.
1. Ship your bike from home about $1000 to $1200 round trip.
2. Buy a bike in Copenhagen and ditch it in Berlin. $600 to $1200 depending on bike quality. I would suggest donating it to charity and taking a tax deduction.
3. My idea. Fly to Berlin, rent bikes, travel to Copenhagan on Bike eligible trains. Best would take 9 hours and have 4 changes. Search at http://reiseauskunft.bahn.de/bin/query.exe/en?country=USA. There is a box to check at the bottom of the page to select trains you can bring bikes on.
4. Rent bikes in Copenhagen and have them shipped back from Berlin. Above website suggests:
https://www.myhermes.de/wps/portal/paket/Home/privatkunden/preise/nationaler-versand/fahrradversand
This might require you to hang around a few days to confirm shipment is completed. Add insurance.

I have done a lot of cycling and I have never seen a rental where you could drop it in another city similar to car hires. That would required some type of a network of bike rentals to move bikes around. I am guessing you are out of luck. Our solution now is a Bike Friday. When done, fold it into the suitcase , and carry it home. Have solved the problems of airplanes and trains that do not want to take bikes. You might call a couple of shops in Copenhagen and see if they have a recommendation.

Good luck. Would love to know how this turns out.

I would seriously consider bringing your own bike with you. Copenhagen to Berlin is some serious distance. I have never rented a bike that comes close to my touring bike at home, plus there is the added benefit of having a bike that you are comfortable on, seat that is broken in, and knowing what your bike will do in a given situation. How are you planning on transporting your gear? If you go for a package tour they might have rental bikes and haul your stuff, otherwise you will need to make sure your bags are compatible with the rack.
You will have to run the numbers, but every time I have looked at flying my own versus renting the rental is more expensive than bringing my own bike. You could either ship your bike as a parcel or through your plane ticket.

We've toured in the USA and brought our bikes with us by car, but for self-guided bicycle tours in Austria, France, Belgium, and Ireland, we've always rented a bike over there. We've had our own panniers, and some outfits let you swap out their pedals for yours, but having a bike computer (if that's important) is less available, unless you use a smartphone or install one on your rental. We've always stayed within the same country, however, but the Austrian train stations let you rent a bike from one station and turn it in at another station, and the company we used for Ireland shipped their bikes and picked them up anywhere in Ireland for about €25.

One other option to add to Sam's possibilities would be to have a Berlin bike rental outfit ship your rental bikes to Copenhagen, then you simply turn them back in when you reach Berlin. You'd be on your own to get the bikes reassembled out of the shipping carton and properly fitted (maybe including realigning the handlebars, reinstalling the pedals, and repositioning the seat height, along with mounting the wheels), or they could have the bikes shipped to a Copenhagen bike shop that could do that work for you.

Sounds like a fun trip. Whether you hire a touring company or set out on your own, enjoy!