My dog passed away just short of her 19th birthday, which is very geriatric in dog terms. I believe she would have been the equivalent age of a human in their 100s. She was blind, deaf and arthritic, but still quite vibrant until her last few weeks. She needed help to move around and was the topic of a great dispute at home regarding how much you could manoeuvre her around on Shabbat and holidays to help with her feeding, drinking water and bodily functions. To this day, I will regret being told I could only just push her with a foot or some other method to move her and regret not being allowed to pick her up and cuddle her on Shabbat and holidays in her last failing year. None of the activities were "playing" but in the realm of "caring for" and "comforting." Blind and deaf, she needed to be touched to feel safe. What would be the halachah in this instance?
Igros Moshe Orach Chaim 5:22 holds that pets are not Muktzah at all. This would mean that even if the pet is feeling fine you may pick up and cuddle your dog on Shabbos. Although there are those who disagree with this ruling, possibly in a case of hardship, pain or discomfort your rabbi will permit relying on it.
Yabia Omer 5:26 says that one may move a household pet to alleviate its suffering.