You are getting a little mixed up here.
To be prosecuted for an Overload offence, the suspect vehicle or outfit has to be weighed, and one or more weight limits must have been exceeded. If no weight limits have been exceeded they cannot prosecute an overload offence as no "Overload" has occurred.
Applying that logic to your scenario - where a car cannot pull more than 1600kg , the measured weight on the trailers road wheel must not exceed 1600kg. The MTPLM is a weight limit or capability, it is not a measured weight, so provided the trailer is only part loaded and does not exceed 1600kg on its road wheels an overweight offence has not been committed.
However driving licences are different, and that is where the mathematical sum of the tow vehicle and trailers MAM would be used and no measurement is necessary - that would result in a no licence to drive the vehicle offence not an "overload" offence. But this is an area the Govn't has vowed to change- see other threads.