I have been spending my spring vacations riding the Arizona strip area since the late 1970s. Something to keep in mind when you ride south from St George towards the Grand Canyon...in general, the further south you go the higher the elevation. Just the opposite of what most people expect. Middle of April, don't be surprised to find a lot of snow and mud. I generally go down there about the middle of May. Two years ago when heading for Oak Grove (jumping off area for Twin Point, Kelly Point, etc., I had to turn back because of mud and snow. Last year - no problem. Roads were dry and dusty for the most part. Same if you try to get over the shoulder of Mt Trumbull to get to Toroweap - you're looking at riding at 7000 feet and mid-April the roads may not yet be passable. You'll have to check locally for what it's like this year. On the other hand, both the Oak Grove and Mt Trumbull area are great for camping if the snow is melted off - giant stands of Ponderosa pines for shade and to break the force of the wind, nice and open between trees.
Regarding the road to Kelly Point - be aware that the last 20 miles or so, your butt and the seat on your KLR will not spend much time bonding. It is rough! You'll be up on the pegs most of the time, bouncing from rock to rock. I've done these roads on everything from an XR250 up through a 600 and the smaller bikes were much easier to handle.
If when you get to the St George area you find that there is still deep snow in the higher elevations of the Strip, start from Colorado City and head down toward Toroweap down Clayhole Valley road where the elevation won't be as high. From there you can check out TuckUp Point and SB Point also depending on how the roads are.
The Yamaha, last year on Snap Point: