You're right though, death of a pet is inevitable, but your writing made me think about what I would do with Muppet (my dog's name) given the injuries and suffering that was described to me. I was thinking well, I don't want to take him to a library because I want to spend time with him doing something he'd enjoy but the options made me go there.
I'm not clever or a good writer but perhaps there's some sort of a balance between drawing it out and not making it a straight cut to the end. And don't apologise for making me upset, that goes to show that you're on the right track and you're successful at your goal!