Let go of your animosity. Obviously you and your ex have different expectations for what is best for your son. It is what it is and you will be dealing with this for the rest of your son's life with you. Fighting with your ex will only get worse over time.
Would you rather be right or be happy? This is a question I ask myself every time I find myself in a fight. You say that going is not bad for him so take him. Not going to prove a point is bad for both your health and your son's well-being.
I suggest that if you were to just agree with your ex and his mother they will have no reason to bug you. Life will go on in peace. This is a good issue with which you can learn how to parent your child along with input from your ex. As his father he has the right to being involved in deciding what is best for the son you share with him. Just accept that as a given and go from there.
Do not assume that they are being critical of your parenting. Also do not think of two mornings a week at a Mother's Morning Out as daycare. It is an enrichment program designed to give mothers an opportunity to do things on their own. I suspect that if you'd been a stay-at-home mother for longer than 2 months you would welcome this opportunity.
You have turned a reasonable request into a power struggle. Have you considered the possibility that they made the suggestion believing that this would be helpful for you? If you voluntarily left your job you're not job hunting but do you not have any activities in which you'd like to participate as an adult without having to first consider the needs of your child?
You can change this into a positive for both yourself and your child or you can continue to fight which will cause harm to everyone involved but especially for your son. Your son needs two parents who get along even tho they don't live together. You and your ex have many more years during which you'll have to relate with each other. Wouldn't you rather those years be happy than contentious. You have the ability to make a very positive contribution to a healthy relationship that will be good for your son as well as yourself.
An added thought. It is true that you are not required to listen to your mil or do what she asks. If she were the only one asking I'd literally not listen to her. And I definitely would not let her upset me. But you say that your ex also wants him to go. Even if you do not share custody he still has the legal right to be involved in parenting his son. Visitation is no longer called visitation. It's called parenting time.
It sounds immature for your ex to fight his battles thru his mother. However, you can't change him. Perhaps this is at least part of the reason he is your ex. So accept what is and work with it instead of against it. If you can, in a mature manner, tell your mil to not call you because you will only respond to your ex's requests tell her to stop calling. Then calmly tell her you're hanging up when she calls.
You can choose to not co-operate with your ex. Unless he files for a change in custody arrangements you will not suffer legal consequences. But you will make your life and the life of your son miserable. Consider what you can do that will be best for your son and his relationship with his father. For him to grow up emotionally healthy he NEEDS a good relationship with both parents. How can that happen when those parents are constantly fighting with each other?