I don't care that my family's dog will be dead in a few months
Why don't you care about the dog dying?
I just don't see the point of caring about it for several months before it dies.
Even if you don't care much about the dog, you should be there for your family. They may feel very sad about the dog's impending death, so you should be sensitive to that and don't tell them you don't care.
Why do I have to be sensitive about it?
You should be sensitive because it's your family who is being effected!  Even if it doesn't bother you personally, it clearly is bothering them.