"I arrive in Frankfurt on a Monday morning, have 4 days (Mon-Thurs) to play"
"Then I have Wednesday and Thursday (just two days) before I leave from Frankfurt on Friday morning."
It would make sense to visit your more distant target (Bavaria) starting on Monday. Visit the Middle Rhine valley on Wed and Thu.
I like the Black Forest very much, but I think with your limited time that Tom is right - the Neckar Valley offers similar charms. You might have time from Friday through Tuesday while you're in Heidelberg to see some things right nearby. In addition to Tom's suggestions look into LADENBURG (about 15 train minutes from Heidelberg) and Bad Wimpfen. (Triberg by the way, except for the Black Forest Museum, is an oversold disappointment.)
And talk about oversold... Neuschwanstein - King Ludwig's most well known "castle" - isn't a castle at all, just a late 19th century palace/home that looks amazing on glossy tourist brochures. I wouldn't waste the 5 hours it would take you to get there and back from Munich. I'd suggest instead that when you visit Munich, you see Nymphenburg Palace instead. And later, see some real castles in the Middle Rhine Valley.
BAVARIA: Consider a FRA - Würzburg (northern end of the Romantic Road) - Nuremberg - Munich route for Mon - Thu. On Monday you could take the train straight to Würzburg (northern Romantic Road city,) drop bags in a locker, sightsee there, then travel one more hour to Nuremberg for 3 nights. Spend one day on a day trip to Munich. Spend a 2nd day on a daytrip to Bamberg or IPHOFEN + Nuremberg (these 2 daytrips are very doable - only 45 min by direct train each way.) Then on Thursday see more in Nuremberg before you head to Heidelberg. Daypasses for these day trips are very cheap.
MIDDLE RHINE: Wed. morning, take the train to whatever town you'll stay in. (Boppard is excellent - lots of accommodations choices - and would position you well for trips to all the Rhine towns and for the longer trip to Burg Eltz. You can also catch direct trains from Boppard to FRA on Friday. St. Goar is also a good central option.)
Instead of using the train only, it is possible, if your choice of town is Boppard or St. Goar, to get off the train in Bingen with your bags, walk to the KD dock, and board a cruise boat to either town. Boppard takes 2.3 hours, St. Goar takes 1.5 hours, and you'll see the best part of the river scenery with either as a goal.
I would agree with Tom that Burg Eltz would be very time consuming. It might be best to visit the excellent and never-destroyed MARKSBURG CASTLE instead, which you can reach more quickly from the Rhine towns. From St. Goar (home of Rheinfels Castle ruins, by the way,) just ferry across the river (runs all day) and take the train 20 min. north to Braubach. From Boppard, ferry across the river, walk north maybe 15 min. to Filsen, and catch the train 7 min. north to Braubach. English tours at 1:00 and 4:00. Day trips by train here are quite cheap as well - if a little confusing to the first-time visitor.
Rhine/Mosel day passes