She wanted to keep having the dog, but eventually she could no longer care for it. This is the most natural form. You have not introduced the idea that the dog is female, so "it" is the proper choice.
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.
hrsaneiI would write it as; She wanted to keep the dog, but eventually she stopped caring for her/it.But that rewrite would change the meaning considerably.