A big element of Judaism is to keep things in proper perspective. Our purpose in this life is to do God’s will. While God’s will may be different for different individuals, we should still keep that goal in mind no matter what we are doing – “What does Hashem want me to do now?”
Once we realize this, we can better understand the place of pets in Jewish life. Pets provide company, help children learn responsibility, and can be emotionally supportive of their owners, even though they don’t talk. Some pets also aid the handicapped or provide security. These things are all important and help a person lead a happy life. On the other hand, pets are often costly, may cause halachic problems (problems relating to Jewish law), and can be very time consuming responsibilities.
The question is, are there alternate methods you can derive the benefits that pets provide, without having to sacrifice with the negative aspects of owning a pet? It might very well be that there DO exist better alternatives, but detaching from a pet is too emotionally straining to make it worth it. That’s fine.
Once you recognise that the true purpose of a pet is to provide you with various things to help you better serve Hashem, you should no longer feel a need to pray for your pet – you can just pray for the satisfaction, joy or other good that your pet provides for you. Instead of praying for Cleocatra to live a long life, pray that you should live a long life that is free of sorrow and pain. Hashem is all-powerful and can make you genuinely happy with or without your pet. A pet is just a means to an end.
In general, the Torah’s philosophy is to not allow ourselves to become too attached to anything physical in this temporary life. I hope this helps.