WIBTA If I tell my husband that his parents dog is not his responsibility anymore?

My (25F) husband (27M) and I have been married for almost a year. Before getting married he used to live with his parents and they had a French poodle, called Mimi, he used to be the main caregiver for her, but my FIL adores her and he says it's his dog. When we got engaged it was agreed that they'd keep the dog because we moved into a small apartment and are out for work around 12 hrs. They live in a big house with patio and a small backyard and my MIL is a housewife, so Mimi would be more comfortable there. It was also agreed that H would contribute to some of the expenses for her. It was all settled.

However there have been some incidents regarding Mimi that made me feel kind of uncomfortable. I'll give some examples.

1. A few months ago we had plans of visiting my parents (we don't see them as often as we see his parents because they live in a different city) that morning MIL called H and told him that they were going fishing that day and that H had to go to their home to walk Mimi. He accepted (without asking me) and we had to go and were like 2 hours late to my family reunion.

2. We agreed to babysit Mimi for their vacation, when they picked her up they didn't even say thank you. Even though we could not go out on those days because she needs to be walked 4 times a day or she cries.

3. MIL constantly says that it is H dog and we must take her with us. She usually says it as a joke but is way to often. FIL usually says she's his dog and he won't allow us to take her.

4. Lastly, H had been insisting on have her sterilised (he would pay for it), they were reluctant for months and then saying they did not have time. Now MIL that it should be done now (we're still in lockdown here). I feel she is using it to make us visit them (she's been "skeptical" about the virus and says we don't want to visit).

I know H is thinking on picking Mimi up on Saturday to take her to the vet but he hasn't told me anything. I'm thinking that this might be the opportunity to tell him that Mimi is no longer his dog and that is not his obligation to go out of his way everytime to take care of her. I wouldn't mind if his mother didn't act so entitled, so I think there should be a limit. Would that make me an AH?

Edit: I think I did not express it properly. My problem is not about my husband taking care of Mimi, I have no issue with that whatsoever. My problem is MIL dictating when he has to take care of her without considering our plans and my husband accepting everything without even consulting or considering me.