The route you've picked is fine. SR 18 is the shortest way to St. George.
As for Cave Lake, love it. The park has two campgrounds; Lake View and Elk Flats. Lake View is quite tight for larger RVs, anything much over 30' will have some difficulty negotiating some of the turns although I've seen 35'+ rigs in there. Elk Flats is a "newer" CG and better suited for bigger stuff. Of note, Elk Flats has only back in sites.
The campground(s) are primitive but that's more than made up for by the setting. We truly enjoy it! We drive right through Ely and it's couple of RV Parks to stay at Cave Lake.