This may sound too simplistic but, If I were you I would install a new set of those Lasertec tires you were running in the past with no problem.
That should solve your problem.
That weaving, wobbles at speed is dangerous, not only to you but anyone else in the area.
Trying to fix / change a bike's handling characteristics that has tires on it that were not designed for it [ radials ] sounds like a pain, and turns you into a full on test pilot.
Nick, I did save my Lasertecs that were installed last fall and they had less than 1,000 miles on them.
I thought it would be fun to see how these Conti Radials felt combined with the Race Tec Gold Valve Emulators in the mountains. I noticed a big difference in the ride and feel of the front suspension after the installation of the Gold Valve Emulators.
I put about 3,000 miles a year on Conti Road Bike tires on my Specialized Tarmac, and love the way they work after trying numerous other brands. I was hoping for the same results on my R100CS.
Being a Test Pilot is not my idea of fun, even after flying planes my whole life!
Thanks to all who have responded to my questions on the swaying/weaving. Today I had the bike looked over by my local mechanic who only works on BMW's. Checked everything from beads on tires, rear swing bar, steering head etc. All checked out great. Even spun up the rear tire and all looked good. Had the front pressure at 33 lbs and the rear at 36 lbs. He took the bike for a run at 80 mph with no side bags and all felt good. I then took the bike up to 95 and all felt good, again with no side bags. The result is with a little lower pressure and no side bags all is good. Next I will put the bags on the bike to see If that makes any difference.
FWIW there's a good chance you can find in your owners manual that when saddlebags are fitted, speed should be limited to 80mph.
It's in my G/S manual and on a decal inside the lids of the old Denfelds that I saved when my RS moved on.