I think it was more than a little insensitive for you and your mother to visit your SIL almost immediately after the death of her father when your relationship has been strained (at best). Saying it came from “love” seems OTT as you do not have a loving relationship with your SIL. In her position, I can see how it could come across as more than a little false and intrusive. I don’t think shouting at you was necessary, but people don’t always act 100% rationally when grieving. I’d chalk that down to misadventure and lack of thought on your part OP.
As to the wedding, I think it is fine to not invite children, including very young babies, to a wedding- as long as you fully accept that many people with children will decide (or have to) decline the invitation.
Equally, if you don’t like the way someone is organising their wedding, then you simply decline the invitation and move on. Obviously, if the reason is you can’t/don’t want to leave your child then you simply say why and I would expect there to be complete understanding, given the circumstances. As it was your siblings wedding I think it ok to say- “ok but you realise this means I can’t come, even though I’d love to be there”.
What you don’t do is kick up a fuss until you get your own way. Or get parents involved to agitate on your behalf until you get your own way. It won’t turn out well in most cases- even if you get your own way in the end- you are still resentful re the original slight/problem and they are resentful that they were manipulated into doing something they didn’t want at their wedding.
Your father’s behaviour was pretty appalling in my view- a gift should be freely given or not given at all. Or the strings attached very explicitly stated from the very start, so the receiver is in no doubt as to what they are agreeing to. Anything else is manipulative and unfair, usually leads to further resentment down the line and it rarely ends well. To then try to reinstate the “gift” when he got his own way is clumsy, at best, and I’m not surprised the father of the bride declined. I’m not sure why your father is upset, though- he thinks he did the right thing, after all.
Regardless of whether you agree that OP’s DB/SIL have a right to exclude babies from their wedding, it is blindingly obvious that they did not want the baby at the wedding, were then pressured into it- including being manipulated financially by OP’s father. It’s not exactly surprising that OP’s DB and/or SIL were not have been entirely happy at the outcome and are resentful about it all. Others may not agree with what they did, or their reasoning, but the outcome was that they were coerced into having something they didn’t want on their wedding day. It’s almost certainly going to leave some mark on the relationships going forward. And relations have indeed been strained since- not surprising.
OP and her father got what they wanted in the short-term (child at the wedding) but at the expense of causing resentment both ways and strained relationships. I’m not saying DB and SIL are “innocent”, as I don’t think either side handled it well or covered themselves in glory, by the sounds of it but I do think the bride and groom should get to decide on a child free wedding, if that’s what they want.