Volvo (for example) fit larger oil coolers when fitting towbars, to stop the automatic gearboxes from burning out. So there is a potential problem there, for the gearbox cooling and for a larger radiator needed to keep the engine cooled properly.

I don't see how a larger petrol tank would keep the gearbox cooled, however.

So yes, if you just fit a towbar and start towing to the max weight, then you may well get problems. Or you might not.

But it's a 9 year old vehicle, and you might well get problems anyway !

Russian roulette, really.

If I was going to start towing with a vehicle like that, I'd probably just get a towbar fitted by Witter (or equivalent), and make sure the car was very well maintained. That would include a gearbox flush and oil change.

A number of German cars get additional equipment when a factory-fit towbar is ordered - at original build time the additional cost is reasonable but to change those afterwards would cost a fortune.

The good news is that the engineering side of the manufacturer (not the dealer) may allow a lower towing limit if the additional cooling isn't fitted.

My own car, a VW Touareg, has a 3,500kg towing limit with the factory-fit towbar and cooling but "only" 2,200kg without the additional cooling - perversely only Touaregs for UK and Northern Europe have the smaller fan as standard - Southern Europe, Australia and North America all getting the larger fan as standard so no problem with afterfit there.

If you can contact M-B in Germany, or M-B in the UK failing that, find out if they'll advise a lower towing limit without the extra cooling.