I have the same issue...I'd like to take my BMW R1150RT, about the same size & weight as your GS. Towing another trailer behind the fiver is about the only real solution other than a toy hauler. I don't think it's legal in FL, tho. I was thinking of getting a smaller lighter bike that I could haul in a carrier mounted in front.
2008 Ford F250 Super Duty, Lariat, 6.4l PSD, Super Cab. Long Bed
2006 Jayco Jay Flight 27.5 RLS Fifth Wheel