Mitsubishi Pajero 2013: Should Mitsubishi cover the cost of the fan belt replacement?

Driving on Fraser Island the fan belt on my 2013 Mitsubishi Pajero failed leaving me without an alternator and water pump. I contacted my dealer and was questioned as to why it had failed. They said the warranty would cover the cost of the part and labour to install. When I returned and presented the bill to the dealer they said they could not cover it because the island mechanic had charged me too much, and it was a simple job to change a belt. I believe they should pay the total cost plus the cost of installing a genuine belt to replace the one fitted on the island and replace the coolant that was lost. Am I expecting too much?