We would like to stay 3-4 nights in Bavaria before traveling to
Budapest, October 3 or 4 for 3 or 4 nights... Would like to revisit
Munich, but not a deal-breaker. Would like to see the Fuessen castles,
Rhine River trip (KD lines, I presume?) and would like to revisit
Rothenberg...
It's very helpful to have the travel context you provided.
Sounds like you and I were taking our first eurail journeys around the continent at the same time.
While the Rhine Valley and the nearby Mosel Valley together constitute the very best part of Germany, IMHO of course, this area is not in Bavaria and is nowhere near Munich, Rothenburg or Füssen.
This map shows the location of the Rhine & Mosel rivers (look for Koblenz) and the scenic train rides (and cruises) that are available there. Castles - real medieval castles like Burg Eltz & Marksburg - are found in this area and offer great 1-hour tours. Stunningly handsome old-world villages like Bacharach and Cochem are easy to access by local train, full of half-timbered buildings like Rothenburg's, and dotted with wineries as well. Some towns have chairlifts or gondola rides that take you to the clifftops for awesome views. The natural scenery is quite stunning. We have based ourselves for 3-4 nights in the small riverfront towns, where views like this one in St Goar can be enjoyed over a few days instead of for the few minutes your cruise lasts (This was taken from the Rheinhotel St Goar, where a "double room large with Rhein view and balcony" includes breakfast and goes for €140/night. Lesser rooms are available at lower prices. See the KD cruise flag in that photo? You are close to the cruise dock without question here - and close to the rail station as well. https://www.rheinhotel-st-goar.de/en/prices/ ) Hotel Rheinfels, next door, enjoys a similar location. Other hotels in these towns go for lower prices and can be quite nice.
If you go to the Rhine - where the castles really are real and really have been there for close to 1,000 years... you will probably not have time for the Füssen-area "castles" (Neuschwanstein is a much more modern building with a fake castle exterior) although you might have time to stop in Munich on the way to Budapest. There's plenty to keep you busy on the Rhine for that short amount of time. I sense however that this is more of a "nostalgia" trip and that you are less interested in visiting new places and more interested in returning to places you have already been. And in this case, I suggest you consider Augsburg. The Hotel Ost am Kö is a wonderful little hotel with good prices. I spent just a night there this summer. Augsburg is a nice enough town and is close enough to Munich, but only slightly closer to Füssen