Dave: I think you are correct that this is a nice walk and generally doable.
That said... though weather and trail problems MIGHT be avoidable in April, there's no guarantee (nor any guarantee that the kids are up for longer walks.) But beyond that, the unavoidable problem with this family's walk to Eltz and back is one of time, and fitting the castle visit in with the larger plan for the area. From a Bacharach base, the OP has 4 hours of train travel time altogether wrapped up in Bacharach > Moselkern > Bacharach. Add 2-3 hours round-trip walking time for the castle trek. Leaving Bacharach as early as 8:05, they cannot possibly make it to the castle before 11 am. Add lunch time somewhere. "Perfect timing" for arrival at Eltz Castle for tickets and the 1-hr. tour, and for catching a train at Moselkern station after the walk back down would be advantageous, but they will likely have some "wait time" in both spots, so add time for that too.
I've walked to Eltz on separate occasions, both times starting out by train from Cochem, which is only a few minutes from Moselkern by direct train. It worked fine - I had a relaxed time with some afternoon time left over after the castle tour for other things. When you did this walk, were you based in Bacharach, as the OP plans to do? If so, you probably found it tricky to fit in anything else. And so will this family. From Bacharach, the OP hopes to have a Rhine visit and a Mosel visit (exploring towns on the way) as well as a Rheinfels Castle AND Eltz Castle in 2 days' time. I just don't see how it all fits. Something needs to go.
Rick promotes Burg Eltz as his "favorite castle in all of Europe" in his article, "Slowing Down in Germany’s Mosel Valley. I suspect his praise for Eltz is the reason that so many forum posters feel compelled to take the tour there. And it really is a nice castle to visit. However, Rick's own 21-day tour of Europe, which offers a 2-night stay in Bacharach, does not include Burg Eltz - probably because it's just too problematic. Note also Rick's title, a clear clue to readers that the Mosel Valley is tricky on blitz-tour visit.