This is characteristic of the lightnings, short wheel base and rider position.
Just to be sure, always check for wheel bearings failures, swing arm bearing failure, steering head bearings, and engine mounting. Engine mounts at front sub frame, under air box frame to engine. Remember, the rear swing arm is solid to the engine, not the frame, if they arent secured to eachother, means tank slappers.
What you describe sounds like youre in need of a steering stabilizer. My lightning, and CR both get this kinda wobble