I think that since this woman is about to be a very intricate part of your family that you should probably make an effort to involve her and spend time with her. I also think that since she is essentially family and your mother is family, that the two of them should be able to get to know each other better. Your FMIL will never warm up to your mom if she doesn't have the opportunity to spend time with her.
However, it is reasonable to want some alone time with your mom as well. Perhaps you need to discuss this with your fiance and come to a compromise. Maybe your FMIL would be more than welcome at half of the days events, but you and your mom can have some alone time just the two of you (without your fiance as well). Your fiance may feel that if he's there, it automatically becomes family time, and that's why he wants his mom to be a part of it too. Or maybe he is uncomfortable with your mom (that kind of shyness can run in the family) and wants his own mother there as sort of a security blanket.
Whatever the case is, you need to explain your feelings clearly to your fiance, and ask him clearly why he wants his mom with for everything. Do not use metaphors or big words when you're explaining what you want. Make it short and simple. Say "I don't get a lot of chances to spend time with my mom, so I'd like to have some alone time with her this weekend. I want to involve your mom in activities, but I'd also like to leave some for just me and my mom. What activities are important to you for you and your mom to be there? We need to decide which activities will be all of us, and which ones will be just me and my mom." Bam. Clear and simple. No room for guess work. It makes it clear that you intend to spend quality time alone with your mom. It opens up room to discuss what's important to him so you can come to a compromise.
From what I've read, it doesn't sound to me like your fiance understands that you want mother daughter time. He told his mother you didn't, so he must think you don't (or he just doesn't respect your wishes, in which case, please don't marry him!) You can't expect him to just know this. Do you just know why he wants his mother there? No, you don't. And while we're on that subject, maybe you should ask him. "Is there a reason why you want your mother with you all day on Saturday?" That's the only way you're going to find out.
Tell him clearly "I want alone time with my mom" or maybe "I want my mom to get to know you better through some quality bonding, so lets schedule one event with just you, me and her". Tell him what you want to do and why. Don't expect him to know it.
And while we're on the subject of clear and open communication, have you considered calling his mom and asking her if she wants to come or not? She might be a lot more understanding about this situation, having been through two weddings herself, than your fiance is, since he's likely never done this before. When you get married, his mother isn't going away. You need to establish a good relationship with her if you want to have a happy family life overall. It's not cool to force your husband to pick between you and his mom because you don't feel like talking to her.
Why don't you try actually talking to your future husband and future mother in law, instead of talking about them? It's not really your fault that talking about people rather than to them is your first instinct, it is what we are taught by society. Women aren't really taught communication skills, we're taught all this codependent nonsense that leaves us saying "I don't know what to do, other people are doing things I don't want them to do, and they don't understand why they shouldn't." Well, first of all, they probably don't even know what you want, because you're not telling them. You don't know what they want, because you're not asking them (and in many cases, you might not have thought to care about what they want). And none of you are able to come to a compromise that makes all of you happy, because none of you are talking to each other.
The solution here is to talk to your fiance and your FMIL. Now is the time to learn how to communicate effectivley and respectfully with your fiance, because it's not going to magicallly happen once you're married, and if you can't communicate effectively and with respect, you might as well not even bother, because the marriage is not going to last. Nothing magic happens during a ceremony to make you both suddenly able to read each others minds and agree on everything. Marriage takes work, communication, and compromise. Theh sooner you both start doing that, the easier and happier your marriage is going to be.
Good luck and congrats on your engagement!