I believe that the children exist, they are his and hers and she carried them.
From what we have seen from Harry and Peg, they are way too stupid to pull off an elaborate ruse like that.
The belly was shaking and moving weirdly because just like all of her underwear, she chose the wrong maternity shapewear. Not every woman has a smooth and round belly, some children move a lot in the belly and a knee or elbow will stick out to make it look incredibly weird. Also, the baby can move into different shapes which affect the size of the belly, within minutes. I think she was pregnant.
Archie's birth was murky because they wanted control. They wanted to teach the British media a lesson because they were calling her out at every second. They wanted the US Media to be the first to announce his birth but the royal family pulled a fast one on them. In the end, they ruined their own experience. Harry was high on gas, Peg was in pain, stressed and allegedly suicidal, all the while they were trying to outmanoeuvre the media instead of focusing on the birth of their first child.
They think they got the media? Nope. The media lives inside their heads. For free.
There are Baptism certificates though. I have one.
It has my name, my parents' names, my godparents' names, the church I was baptised at, the date and the Priest who baptised me.
People in the UK don't get certificates?