One thing I found when my son was riding with a tag-along was that the pivot where it attaches to the seat post needs to pivot freely. When I first installed it I had tightened it to the point where it was almost a rigid connection. Riding this way made the whole vehicle really, really wobbly and unstable. Once I figured out that it needed to be freely pivoting it solved the problem. I'd bought it used so there were no instructions, and probably some missing components to the attachment.
My son wasn't quite as heavy as yours, but you might still want to check the pivot attachment.
Good luck.