Okay, so they wanted (and booked) a child free wedding. They had to adapt this to accommodate not only your children, but your cousin's children and the BILs children. So there are an extra 5 guests (assuming the baby was babe in arms, usually not counted in a headcount and the cousin also had 2 kids).
So let's just say that the price per adult was £55 for the breakfast and therefore children may have been discounted down to £35. So an extra £175 is no small amount really considering a 2yo and 4yo probably aren't going to eat the full meal and behave themselves.
I've worked in the wedding industry for over 10 years and I actually dislike working a wedding where children are invited (usually) and I wholeheartedly agree with child free weddings because:
Most venues have a maximum capacity. They can't go above this for fire regulations and space. I'm not sure how many of you have carried a red hot plate from a regen oven and had to squash your way around a very crowded breakfast room where you can hardly fit between the chairs, but it fricken hurts and you're filled with the fear that you're just going to let go of the plate to save your burning hand. If the B&G had planned to invite say 50 guests and reached capacity, they then would have needed to find a way to accommodate an extra 5 (highchairs do count btw) so this may have been either a) disinvite some of their friends/extended family members b) pop the kids with a parent each into a side room. Seems they made the fairest choice. The alcohol issue may have been that that room is not licenced. If that is the case, there is no way you would get a venue to break their licencing agreement and possibly face a fine, a loss of liquor licence, individual staff fines and immediate dismissals.
On top of that, the waste from kids is much higher, on average they don't eat the full meal, they make a mess, ruin chair covers and table linen which perhaps the B&G wanted to reuse for the nighttime (instead of paying for it again).
Finally, children are messy and loud. I've seen so many speeches ruined by kids running around, knocking over wine, shouting, watching iPads etc. Yes even the well behaved ones will ask to go to the loo just as the bride's says "I will" or when FIL is getting emotional in his speech. I've had a very close shave when I was serving very hot soup in a very hot plate and a young child decided to run up to me and tug on my dress causing me to spill the soup. If it had of landed on them, they would have gone to hospital (yes food does come out of the kitchen red hot if you have any distance to walk). Kids also usually bring toys which are scattered around the table and the floor, making it difficult to serve.
Yes I love it when my DS is invited places, and I got weepy when he wasn't invited to a recent wedding, but purely because I liked the thought of dressing him up and taking cute pictures. Bride's wouldn't have gained anything from having him there, in reality, I would have been running around after him all night and early to bed so missed the reception, he wouldn't have liked being stuffed in a suit and told to be quiet and still...basically just no.