Don't be a reductionist. If that's all your cat was and that's all your relationship was, then fine. If it wasn't, you owe it to that cat to remember all the good times you two had between when it came to life and when it lost its life. Do it for the cat.

Then, when you've finally moved past this point, realize how much joy you gave that cat. Know that you did something beautiful in letting that cat know it was loved from beginning to end. Think about the cat's perspective. Its death may have sucked; they rarely don't. Now think about every moment that cat experienced growing up and being with you. Every sense of relief that cat felt when you came home. Every wave of comfort when you gave it pets. Every moment of safety it felt when you cared for it.

I know how sad you are, and I know this is tough love here, but that's what carried me through losing my dog on Christmas 2022 and her sister December 16th 2023. We did our best. They couldn't have had happier lives. I'm glad we could do that for them. It was worth the weeks of agonizing grief for the 14 and 16 years of happiness they experienced.