Excellent choice for Bardstown. You might want to note that if you are a golfer, the campground is right next to a golf course.
Also on the property of "My Old Kentucky Home State Park" is an Amphitheater that features nightly performances of "The Stephen Foster Story"...a tribute to America's first musical super star.
As mentioned...Bardstown is closer to 5 hours from South Bend. Within the 4 hour limit, you could consider some Southern Indiana State Parks.
O'Bannon Woods State Park is located in Corydon, Indiana, and one of our personal favorites. Lots of trees, great new Aquatic Center, and nice Nature Center. There is a 1850's Settlement recreated, complete with a Hay Press Barn and live working Oxen...Forrest and Gump.
The town of Corydon is the State's first Capital, and the Original Statehouse is located in the downtown area.
There are plenty of caves to explore and wineries in the area. You will also be close to Louisville, and all those attractions mentioned above.
Like most Indiana State Parks, O'Bannon Woods sites are electric only. If that's a deal breaker for you, consider Charlestown State Park. The newest of Indiana's State Parks, Charlestown doesn't have as much tree cover as the other, but does feature some Full Hook up spots... and you'll still be close to all the attractions mentioned.
Another thought would be Versailles State Park. Plenty of tree cover in the campground and very well maintained. There's also a very nice lake for fishing. You'll be close to Cincinnati all that has to offer...Reds Baseball, museums, and the Newport Aquarium.
20 minutes south of Versailles is Madison, IN and Clifty Falls State Park. It to is very nice, with excellent hiking. There is a campground there as well. We've stayed there a few times....some people like...just not our preference. There was only one bathhouse, and all the sites were gravel...though some had hardly any gravel at all, and made rainy days a muddy mess. That's why we prefer Versailles...asphalt parking and multiple bath houses that are well maintained.