A kangaroo cull is unlikely to significantly damage tourism to Australia. The vast majority of tourists will be unaware of the issues surrounding the kangaroo population in Australia, including the cull. Even if tourists were aware, it is doubtful a significant number would object enough to personally boycott the country.
I firmly believe that a culling of Kanagaroos will not stop the damage brought to tourism. I think culling a large portion of the Kangaroos would be very difficult to do. It also seems that the costs would be to exorbitant to stop the onslaught of Kangaroos. They are to capable of survival.