Oh? So the K1600 is supposed to wobble? Your comment is so helpful!
I see a lot of BMW riders put down Harleys. I think it is pretty distasteful. Many of them have never ridden a Harley so they have no point to compare.
I rode Harley bagger for 15 years and abut 200,000 miles. They are solid feeling bikes on the road. Never a wiggle, except for my 2005 that would get a weird sway when ridden over 100 MPH, but as I can not legally ride that sped anywhere in the USA I don't see and issue. I will agree that Harley builds a low horse power, rough riding motorcycle, but the touring frames are decent bikes, just not rocket ships or build for high speed curves. You can ride mountain passes with one reasonably fast and enjoyably although some bikes will do it better.
There has been a lot of comments of unstable K1600s over the years. Some do it, some don't, nobody seems to know why. I have a buddy with a K1600GTLE and it rides dead solid all the way into triple digits. I have had it over 125 MPH myself a few times. If I had just dropped $30K in a new K1600 and it wobbled like some of the videos and info I have seen someone would be buying it back or talking to my attorney.
https://youtu.be/DVxkSxRx6sQ
The bagger is primarily a styling exercise with great marketing potential for conquest sales, but the resulting wobble must have proved difficult for even BMW to fix so they simply compensated with the speed limiter rather than re-engineer the bike.
Like others, I suspect the problem is airflow management and the styling mods unique to that model are to blame. No easy fixes then, unfortunately. Decent looking bike, though.