My non-lynxmotion platform will also just barely turn on carpet, but I have to use almost full throttle to do it. Not very well controllable. Sounds like yours is similar.
Longer wheelbase will definitely make it harder to turn. However, it seems to me that even if one diagonal is lifted, you still have drive from the other 2 wheels, due to being 4WD, so why is that a problem? It should still go. I would think that high-centering may be more of an issue - perched on a hump with all the wheels off the ground.
You might check out the Wild Thumper arrangement too.
Here is my "secret" plan to deal with the turning problem. Install a caster or similar wheel under the rear end, and use a servo to extend or retract it vertically a cm or so. Then, when turning on carpet/etc, it can be lowered and the platform will turn easily via 2WD traction. Way down in the queue after 100 other projects, unfortunately :-).