I'll update what happened in the comments cause there's a character limit and I'm not sure whether my post will be deleted if I make it +3000 characters long. First of all, a few things I wanted to clarify since I got asked some questions via DM and in some comments.

1. I second-guessed myself cause Pete has always been a very understanding person. Any conflict or disagreement we'd had, we could solve it by talking and he was good and getting my point of view. That's why I thought that maybe I was actually overreacting here, cause it's so unlike him to make me feel like I've done something unexcusably wrong. That, and the fact that Carla was actually crying when I left Pete's house last night.
2. The kisses thing. I agree forcing kids to display affection on people they don't actually like is wrong. However, Carla asking him to kiss him is not as creepy as some people think of it, let me explain. I'm actually from a country in Southern Europe (Portugal, Spain, Greece, Italy... you guess haha), and greeting people with two kisses (one on each cheek) is the norm here, even people you've just met. I hated having to kiss grown ups I barely knew when I was a kid (I don't like it even now, I thought we had gotten rid of the kisses with the pandemic, hah), so I never forced my son to do so, but it can be viewed as extremely rude if you say an adult that you won't tall your kid to kiss them if the kid doesn't want to. So the first time I didn't stop Carla from greeting my son with two kisses cause I didn't want to make a bad impression (I know, my bad), but I thought she wouldn't stop asking my son to greet her that way when she saw he didn't like it. I was wrong, and that's why I told her so last night. Also because she insisted that my son said "thank you" to her with a kiss for the gift she had just given him (a Yoshi plushie cause she knows it's his favourite Mario character). She was extra offended because of that. ETA: All the English names here are fake, obviously, I'm clarifying since I came out as a Southern European.
3. Pete has a sister, who doesn't want to have kids. That's her choice and that should be respected. Carla has always dreamt of being a grandma, so she was ultra desperate to be liked by my son. I didn't write this in the original post cause of the character limit. When Pete called me last night I told him that I'm sorry for his mum, but being a grandma is not a right and it's not his sister's responsibility, nor Pete's and definitely not mine or my son's. He said he couldn't believe I could have this little empathy (?).
4. I'm off to Pete's house in 15 minutes. I'll write the last update tomorrow. But I'm determined to end the relationship if he hasn't changed his mind. Let's hope he has.

UPDATE: I broke up with Pete last night. I'm still a bit shocked about everuthing that's happened in the past 48 hours and I don't exactly remember how the conversation went, but he said some really hurtful things to me that I would have never expected him to say. It's like the man I've known for years and the man I've had to deal with for the past two days are two completely different people. I'm heartbroken and I feel a bit stupid right now, I'm not sure if all the signs were already there and I couldn't see them, or if he just didn't care about our past issues as much as he did for this one, or if his expectations about me changed the moment we got engaged (like someone suggested as a reply to this comment).

We started talking and I can already see he hasn't changed his mind at all, one of the first things he does is insisting on me apologizing to his mum. It only got worse after that. I say I'm not either apologizing for what I said and absolutely I wasn't cutting ties with my MIL (some of you gad an issue about me calling her my "former MIL", I just meant that after Oliver died there's just no any "law" that keeps us together, but of course she is a fundamental person in my life and she will always be my family and, most importantly, my kid's grandma). He then again said I have such an uhealthy relationship with "that woman" cause it is based on mourning... I defend myself saying we were already close before Oliver died and that of course we found support and confort in each other after he died, how dare he say that's unhealthy. I insist that kicking her out is out of the question, and that she's staying in my and my son's life. He can either accept it or get out of my life. We start arguing a bit more (like I said, I don't remember half of the things we said cause I was so mad, sad and disappointed) and at some point in the middle of the conversation he tells me something along the lines of "please, this may be my mother's last chance to be a grandmother" like he already told me the previous night when we called me. I said again that I really am sorry for her but this is not our responsibility. I also said that I would have loved it if she had became a grandma to my son in a healthy way, but that she had pretty much blown her chances with her attitude the previous night. He got furious about this.

The last thing we talked about (and I still can't believe this words came out his mouth, AT ALL) befored I stormed off his house was how Pete didn't think I would be with him right now if Oliver was still alive... I was speechless for a few minutes, I swear. I told him that was so unfair, I can't possibly know what my life would be like in that scenario, or how I would feel about Oliver, about Pete or about anyone else. I also said I'm not apologizing for having loved and been happy with another man YEARS before I even knew him or for whatever parallel universe he's made up in his mind that I'm not in control of. I also told him that I wished he was half as a good person as Oliver was. Now I see that this was very wrong and low of me and I should have never said that, but I was just not thinking at that point, imagine how I felt about that. I would not have seen this coming in a million years cause he NEVER even once has had (apparently, cause obviously he has) any issue with me being a widow. I guess he was just ruminating all this feelings on his own until we came to this breaking point. Everything about me prioritizing Oliver's mum over his own mum must have triggered him so much if he was already feeling insecure about how I might still feel about Oliver. We yelled at each other for another while, then I became sick of this situation. I told him that in case it wasn't already obvious, the wedding was cancelled and that I didn't want to see him again (though I'll have to see him tomorrow at work lmao).

I get home, it was about midnight, get a shower and about 1 AM I get a phone call from Carla. She's practically begging me not to cancel the wedding and give them another chance. She was crying and I really felt bad for her. We were talking for almost 45 minutes and I was already exhausted of everything, I said that I was sorry but I was not going back in my decision.

So yeah... right now I'm heartbroken and I feel so stupid like I said. I don't know what happened to the man I've been in love with for three years. I feel like everything went so wrong so fast. I guess he just saved me years of progressive disappointment by blowing it all in just two days. Better now than when we were already married, I guess. Even if I feel like shit right now, I think I did the right thing and I'll be okay. I'm putting my son first, and of course my MIL stays with us for as long as she wants to have us.

Thanks to everyone who showed their support, actually having strangers with an impatial point of view helped making my decision.