I'm not sure why some people are so fixated on staying in a castle. To be an authentic experience, it should be like it was in medieval times, when the only heat was a fireplace and the toilet was an outhouse suspended over the moat.
It you really want to stay in a castle (but one with modern amenities), one of the few never destroyed castles in Germany is in Harburg (Schwaben) on the Romantic Road between Donauwörth and Nördlingen. And it has a hotel on the grounds (I think in one of the original buildings) with double rooms for less than 100€. They have one room for up to 5 people. It rents as a double for 97€/night; I would imagine the fare for 5 people is more.
A thing of concern to me is that the rates given on the website do not say that breakfast is included, and the restaurant does not open for breakfast.
Harburg is reachable by train, but there is no public transportation in town, and the Bahnhof is 1 km (5/8 miles) from the old town under the castle. It's a non-strenuous walk from the Bahnhof into town on a mostly flat road along the Wörnitz river (I did it in 2007, at 63), but it is a bit of a climb up to the castle. Unless you pack light, you might want a car for that part.
If I were doing it, I would take the train to Nördlingen and stay there for several nights. There is much to see in Nördlingen; it's kind of a small, less touristy version of Rothenburg, with a walkable wall, a church tower you can climb to the top of, and lots of Fachwerk buildings, but no Christmas shop or Crime & Punishment Museum. Nördlingen sits inside an ancient meteor impact crater. Our Apollo astronauts trained on the crater walls because the were thought to be similar to the lunar serface. There is a museum in town featuring the crater (Ries).
From Nördlingen, take a partial day trip by train 15 minutes south to Harburg and tour the castle.
You can also take a bus an hour north to another medieval walled city, Dinkelsbühl.