Yes that definitely looks like a Chrysanthemum to me, and I agree if you trim it you will have a huge ridiculous bloom when it gets closer to fall. Leave it alone and just put leaves on it for the winter and it will come back next year.
i want to weigh in on the toxicity of the plant, It is true it will make your dog vomit, or drool like crazy if the dog is wanting to eat some of this plant I suspect your dog might have some sort of gastro issue from bacteria or worms. You might want to send a stool sample to the vet.
The reason i say this is I grow a ton of these on the side of my walk and have noticed that the rabbits leave the plant alone but the groundhogs eat the flowers all the time, they will clear a whole plant at times. Additionally my turtles love these plants. I have never had a problem with the turtles or any of the local wildlife eating these plants. My Sulcata will literally run towards these flowers to eat them. A turtle is not a dog I know.
In Chinese medicine the chrysanthemum is used for teas that have a bit of an anti bacterial property, You drink it if you feel you maybe getting the flu. The flower itself can be used in salads and although smells strong tastes similar to nasturtium. I am not a doctor or a vet, I am just letting you know if the dog is taking a bite now and again there probably is a reason, animals seem to know instinctively about plants.
