Yes you have a problem . . or 2. A simple partial solution to the CG was already proposed i.e. reduce the fuel load. Regarding the balance situation: you've already proposed adding a sidecar to the Triumph , so why not the KLR too? I think I remember seeing some sidecar designs that members have made. A sidecar can be narrower than one for a passenger and it doesn't have to be fancy. There are numerous sidecar Brit bikes for Trials where there is a class. They are simple as all they want is a platform and grab bars for "the Monkey" that could be a starting point for your design.
Good luck to you.