You’re likely to have the opposite of high water Lochsa on the Kern this spring. The snowpack is ~50% of normal this year. If the weather gods smile upon you, you may see the Upper Kern above 1,500 -1,600 where you’ll find plenty of whitewater to suit your fancy. If not, you’ll have the Limestone run (3/4) and the Lickety to run (2/3)
The Lower Kern is running 410 right now. It likely won’t be much above that til later in May/early June. Unless you have a local to lead you down, I’d stick to the Upper.
Camping wise, there tons of free dispersed USFS camping spots on the Upper to park a 30’ trailer. Or Mountain & River Adventures has a nice campground just above the top of Powerhouse Rapid.
The outfitters on the Kern are great folks and can give you good beta on where to go when you get there. You’ll find Whitewater Voyages, Mountain & River Adventures and Sierra South in Kernville. All good river people.