My roommate died from cancer 3 years ago. It never stops being a sad memory, except that the hard pang of the initial shock is gone after some time. I don't feel guilty for no longer feeling that pang, because I know I still wish it hadn't happened and it still marked my life in several ways, so I haven't stopped doing what I privately call "honoring my pain." The usual feel-good advice of forgetting it all and moving on sounds to me as dangerously close to no longer honoring my pain, by which I mean acknowledging that the sad event occurred, and giving it its deserved place in my emotional landscape, but without letting it define my life.
Several of my pets died when I was a kid, and at some point I just sort of integrated the implicit assumption that every new pet would eventually die as well. If I began with that assumption, the actual event would no longer be such a strong shock. I no longer have pets, though.
For some years I had problems with the concept of acceptance. It felt like agreeing to everything that happened, and I just didn't want to give my consent to a series of adverse occurrences that it's not relevant to mention here. Some time afterwards I found somewhere a different definition of acceptance: it's not about agreeing with what happened, but simply no longer pretending that the world is otherwise, which to me sounded like a much healthier attitude. With that in mind, I'm more capable of enjoying the time with my friends while knowing that all living things die.
I don't know whether any of my strategies will work in your situation, but this might: doctors specialized in the treatment of pain distinguish between the physical perception of pain and the emotional experience of suffering. Your dog has no awareness of its impending death; he only knows the physical pain. As strong as the pain may be on a purely physical level, he is spared the existential anguish that worries you. Perhaps making a conscious effort to not project your own emotional experience onto him may make the burden lighter for you.
I hope I haven't said anything insensitive, and preemptively apologize if it sounded that way.
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