Hi, I am so sorry about your cat. I do think it would be good to inform your xh. I would do that about anything serious concerning my children. I no longer think of them as 'ours', although interestingly xh always talks of them as 'ours'.
But I would inform him that the cat has terminal cancer, is in pain, and this is what you are doing. End of story. It is actually more detached imo to tell him than not to.
it isn't guilting him, isn't expecting him to become involved, or do anything. Just give him the information. He might want to see the cat, or he might not, but you have given him information and a choice.
However unreasonable my xh is, I try and treat him as I would an old friend. He doesn't act that way, but that isn't the way I choose to respond to him.