Pipping in to add that the drive from the Stuttgart airport to Ludwigsburg will almost certainly take longer than it should, due to construction and traffic. Supposedly Stuttgart has the worst traffic of any major metro in Germany, which is one of the reason I try to take the train whenever possible if I'm getting out of town.
So with that in mind, I'm behind Russ' idea. That said, having a car does allow for more convenient stops (and to get to Bad Wimpfen you'd have to get off the train and take a bus...) and give you more flexibility. I've both trained and driven through the upper Neckar valley, I think there's much to be said for either way.
Also don't think the 11am Ludwigsburg - 3pm Bad Wimpfen - Katzenbuckel at 5pm or so is very realistic, assuming you're taking the 27 through the valley. That also often backs up because it's a popular road.
In my opinion rushing along the Neckar valley kind of misses the point, really. It's one of the most scenic areas I've seen living in Germany, yet mostly foreign-tourist free. There are so many beautiful towns to stop in from Stuttgart to Heidelberg.
In my opinion, I'd consider trying to free up more time to wander, via car, from town to town on the 27, and leaving Heidelberg as a half-day stop on your way north. Heidelberg is worth seeing, but I don't think it's necessarily a full-day jaunt, and is far more touristy and has less charms than the unique Swabian villages like Bietigheim-Bissingen, Bad Wimpfen, and Hirschorn am Neckar. (Or in the other direction, 10 minutes on the river in the opposite direction from Stuttgart, is Esslingen am Neckar, which somehow manages to evade the notice of American tourists despite being absolutely wonderful).
And as a matter of record, it's somewhat painful to see the idea of having to spend any amount of time in Stuttgart as a punishment. We're a thriving metropolis, home to numerous great festivals, two of the world's premier car museums, great food at charming wine taverns up in the vineyards WITHIN city limits, and so forth. I understand you don't have time to stop in Stuttgart on this trip, but people shouldn't rule it out, especially when arriving from a long flight. It's a great area to get adjusted to the jet lag (Stuttgart has to be one of the easiest cities to navigate by public transit or foot, and due perhaps to the large number of American and foreign workers here, nearly everyone speaks English and is very helpful and used to dealing with foreigners here)