The engine in my 2013 Hyundai i30 Tourer needs replacing, what should I do?

I agree that 136,000km is not a realistic lifespan for a modern car. Especially one that has been as meticulously maintained as yours. Hyundai engines of this vintage are not without their known problems, but in my experience, Hyundai has been pretty good about replacing them when the failures have been the endemic kind.

Your first step should be to by-pass the dealership you've been dealing with and talk directly to Hyundai's customer service department. The dealership, as you've correctly identified, will be more interested in selling you another car, while Hyundai head office is more likely to be interested in keeping you as a long term customer. Ultimately, the car will need to be inspected independently to figure out what went wrong and what caused it.

By the way, if the dealership is right and the 'damage has already been done' by the time the warning light illuminates, what's the point of having a warning light in the first place? The dealership in question has just revealed itself to be ignorant at best with comments like that.

By the way, even though your car has a 2013 build date, the warranty would have started the day it was first registered. Even so, it's well and truly out of warranty, but if the problem is one of the known faults of that engine, Hyundai may offer you a pro rata deal to assist in repairs.