Your main problem is that you aren't living with your spouse and child right now. And you and your spouse have to rely on your mother-in-law to provide assistance caring for your child, and she might treat the child differently than you would, not just in terms of which church she takes the child to, but in other ways as well.
If the only bone of contention between you and your mother-in-law right now is which church she takes the child to, it's not anything you really need to be concerned about at the moment. Your child is only 3 years old, and that child is not going to be significantly influenced by any church service, regardless of which church it's taking place in. When you return home, the matter of which church the child will continue to attend can be resolved between you and your spouse.
If your spouse is siding with your mother-in-law on this issue right now, as a way of avoiding conflict with her, because her help is needed, that may the best way of dealing with it. After you return home, the situation will change, as will your ability to exercise more authority and control over your child's life.
I have no doubt that you and your mother-in-law may have conflicts, of some sort, or butt heads over some other issue, relating to your child's care in the future. She's a part of the child's life and extended family, and she may have her own ideas and ways of caring for and treating your child, and these may differ from yours, or what you would prefer she do. These sorts of conflicts are almost inevitable in any family where a grandparent is a presence in the child's life, and there are better and worse ways of resolving them. The better ways involve rational discussion and negotiated agreements between the parents and grandparents, the worse ways I've seen have led to bitter family rifts and "enforcements" that included forbidding the grandparent(s) to have contact with the child.
So, this isn't just a parenting issue, it's a grand-parenting issue. And that's a matter that you and your spouse may have to address from time to time, but I don't think it's one you should be concerned with right now, over the issue of church attendance. Grandma isn't harming your 3 year old by taking the child to the "wrong" church. What the child is learning right now is simply that grandma goes to that church.
And, after you return home, I'm not sure you should give your child, either overtly or covertly, the message that the church grandma attends is the "wrong" church, unless you plan on raising an intolerant child, and you want that child to disapprove of grandma in that regard. There are not only different churches in this world, there are also temples, synagogues, mosques, etc. with considerable diversity among, and within, all of them. It's fine that you should want your child raised in the belief system and practices of only one of them, but when you start considering the others as "wrong", and letting your child know that, you should consider what else you're teaching your child, and how well you're preparing the child to live in a religiously diverse world.