Vibrations from an out of balance wheel/tire does usually not go away with an increase in speed. I'd go with the suggested steering damper, but I know doodley squat about sidecars.
It's to do with the drag from the undriven sidecar wheel, as the bike accelerates, the sidecar pulls toward the kerb (due to friction on the wheel) and the rider tends to over steer in the opposite direction creating a wobble. Most European bikes up to the 60's had a friction damper in the middle of the forks and screwing down the damper reduces the tendency to oversteer and thus reduce the wobble. Once the machine has accelerated past this point and the sidecars foreward momentum matches the bike, the wobble stops......