After some consideration and questions asked on both this forum and the FJ Cruiser forum, we made a decision to go ahead with the purchase of a 19'. I have no doubts that the FJ will be a suitable TV for this trailer.
Here is a photo I found of an FJ towing a load that is probably a bit heavier than the 19'. (Not that I would do such a thing with mine).
I will know better next spring how an FJ does with the 19'.