I've had both the Crosscheck and the LHT and own a different make (Voodoo Dambala) of 29er mountain bike.
The LHT feels rock solid when loaded down, but the lower bottom bracket and long wheelbase make it less offroad worthy than the Crosscheck. The Crosscheck works fine fully loaded but isn't quite as "solid" as the LHT. The Crosscheck is extemely versatile - I have 2 sets of wheels for mine. One set of fancy road wheels for using it as a road bike, and one set of bomber wheels with either cyclocross tires or touring tires. It can even handle up to 44c tires. I haven't specifically ridden the Karate Monkey, but it would definitely be a much better off road machine. But on the road it would be overly heavy and not very quick steering.
If the bike is going to be primarily used for touring, I'd recommend the LHT.
If you want to do a little of everything - the Crosscheck.
If you are going to do lots of serious mountain biking - the Karate Monkey.