The rating given for a flexible cable assumes in free air and supplying portable appliances and the appliance your supplying is not portable, it is assumed portable means intermittent use, as with a standard kettle, not prolonged use as with an immersion heater, or tumble drier or in your case large lights.
So you are limited to just over 9 meters due to volt drop, and 50% duty cycle would seem reasonable in each hour of use.
However although it seems 1 mm² is not fit for your installation, if it were not lighting you could have a longer length and if it did not supply fixed items you would likely not be using the items for so long, so to supply the flex is not really a problem it is just your use of the flex.
I see many flexes for sale at 1.5 mm² with the typical blue plug which is OK for most uses, but not for caravans where for some reason it says 2.5 mm² and 25 meters long +/- 2 meters from memory, I have never worked out why you can't use a 15 meter long cable? OK where my caravan is parked today the 25 meters allows it to be run around edge of field, but in the main 15 meters would be ample.
So if you ordered a 16A flexible cable to supply fixed lighting then supplier is wrong, but if you asked for a 16A flexible cable then they are within the limits so OK. Same as asking for a 16A outdoor cable or 16A outdoor cable to supply a caravan, it is not the same. With 25 meters it is just within volt drop with 2.5 mm which is likely why it needs to be 2.5 mm.
It would seem they supplied you with what you asked for, but not what you want. It is the old three wishes thing, wording your request is so hard.