Having looked at the Caravan Clubs piece on towing, it mentions an oil cooler as standard on the Honda Jazz. Whilst our old Rover had an oil cooler for the auto box, I have never seen this mentioned on the Jazz forums Can an oil cooler be fitted to a Jazz with CVT? An oil cooler I believe is absolutely essential as two separate people I know have found the hard way how necessary an oil cooler is if towing with auto transmission..
However, a light trailer perhaps might be OK BUT even a light trailer can be loaded with a considerable weight causing stress to the transmission.
More so if we are to continue to get the weather we are experiencing at the moment. 36 degs outside today. I have never seen that in the 25yrs we have lived here. As an aside, this am went out and the coolant light was out in less than 100 yds. The glass roof was too hot to touch.