I took DD to my fathers funeral when she was 2. I also took DS2 who has autism and therefore can not be relied upon to behave.
But in my family funerals are sad but they are not stage managed. It is a family event and being in a family , for us, means accepting that a child may babble or cry and it isn't a problem.
Equally I have brought my children up to recognise me as a human being who is capeable of being silly or sad. I would be horrified at the notion of trying to get my children through their formative years without ever having seen me cry. Had they not seen me distressed aboutthe death of my dad I am not really clear what kind of a message that is meant to be sending?
DD sat on my lap, DS2 read a comic and had an arguemnt with DS1.
When I cried DD sat on my lap and gave me a hug and she toddled down the aisle to go and sit with grandma too - my mum loved that.
All the older grandchildren participated in the service as they had a right to say goodbye just as much as the adults. They went up to the front and read a poem they had written about dad - it was really sweet. My mum was comforted and my dad would have loved it. The view of more distant fa,ily don't matter.
So, OP, I don't think your mothers view matters. The only important views are your MIL , your DH and his siblings.
See what they think and then decide.
BTW DD remembers faintly that day and has kept the dress she wore. It is nice for her that she knows she said goodbye to him even though her memories are sketchy. And she loves chatting with my mum about it, who also enjoys remembering that day as it was a true celebration of my dad.
( unfortunately DS2 didn't get quite as much and, all these years later, still believes that my dad is at Tescos and )