Well I've been to 4 on your list.
Ludwigsburg is unique - if you haven't been to another Baroque palace. I find it quite interesting because historically I've focused my studies on the Wurttembergs and in that context it's great because it's full of interesting Wurttemberg history at the time from them becoming a rather pathetic, provincial power to a major player in European politics (hence the bust of Napoleon).=. It is the best preserved Baroque palace in Southern Germany, and the furniture and fittings are mostly original and unchanged. It is also the site of a dark period of German history, the lynching of the "court Jew Suss" which is worth researching before you go. The tragic story was turned into a horrific propaganda film by the Nazis - there is an American documentary about the family of the director that's available on Neflix. All that aside, the Wurttembergs were relatively poor as far as monarchs go, and if you've seen, say, Versailles, Ludwigsburg will come off as cheap and chintzy and the gardens will be of more interest than the palace. If you haven't spent a lot of time visiting opulent Baroque palaces, however, it may impress you, and the English tour is quite good in my opinion. Also on the tour is one of the oldest continually operating theaters in Europe, if you're into that sort of thing.
Heidelberg schloss is mostly a ruin, but it's a grand one. It's a uniquely beautiful Renaissance castle with a few good legends and historical stories associated with it and well worth the visit, in my opinion. I quite liked the pharmacy museum in the palace as well, even if that sounds nerdy. My advice is to yes, see it, and take the funicular up and walk down, after walking around it.
Burg Eltz is simply stunning. It's what we Americans think of when we think of a quintessential European castle. We visited on a Saturday, while staying in Bacharach, so we took a train and bus up, and did the (easy) hike down instead of hiking both ways. It was simply lovely, although my friend and I had a laugh over RS's description of the forest walk from the castle. (Something about fairies disguised as falling leaves whispering "watch out" while sparrows armed with crossbows...IDK, it's pretty amazing.)
I find it's easy to get castle overload on a trip, though. I'm usually pretty good with one ruin, one preserved true medieval castle, and one baroque palace every six months. Otherwise they all tend to run together. I'd honestly pick one of each and visit them and disregard the rest. Burg Eltz is my "must see" from your list. It's one of the few RS recommendations from his Germany book that lives up to the hype.