We did Neuschwanstein this summer (and 15 years ago) and definitely recommend a car. The English language tours fill up fast so I'd buy your tickets now if you haven't already. Make sure you walk up to the Marienbrucke while you are there for the views. Driving through Switzerland (we did that in 2019 from Paris) won't be a problem at all. Double check if you need to purchase a vignette for the countries in which you plan to drive. We didn't know about it in 2019 and ended up with a fine - I think from driving in Switzerland from France.
If you are into hiking/outdoor activities and have time, the area around Neuschwanstein is beautiful. We stayed in Garmisch so I'm more familiar with that direction. We did a half day bike tour in Munich and everyone enjoyed it more than expected.
Mom to Two Wild and Crazy Boys and One Sweet Baby Girl