Do it (child-free), but definitely don't cite any reason such as space. For the determined you have just given them the key to being an exception: Numbers? - "I know you were worried about numbers but Olivia [who is 4 and huge] will just sit on my lap and I've brought this giant bag of organic Olivia-friendly foods I'll badger your rushed-off-their -feet serving staff to warm and serve." Your premises aren't child friendly and there's open water? - "Don't worry, I won't take my eyes of her." You want everyone to be able to hear the ceremony and speeches and to drink and dance without worrying about children? "Oh Olivia loves a party! She's been to lots of adult events and is a big hit with the other guests, and don't worry, we'll keep her entertained during the speeches."
All reasonable parents will accept it, the tiny minority who don't are unfortunately the very ones that will think it is hilarious and instagram worthy to see their little character licking the icing on the uncut cake, running around the table, wandering up the aisle etc.
Some parents won't be able to attend or won't choose to, which is fine, but in my experience, the fellow guests who have had the most issue with it and are the most vocal, often aren't actually the friends or family who have no childcare or who never leave their children; they are parents who see a wedding as a fantastic opportunity to show off their uniquely entrancing offspring, all dressed up too, to what they deludedly assume is a rapt and grateful intoxicatingly large audience. They often cite the presumed disappointment of others that little Olivia won't be there. Or what a shame it is because Beth will be flying in from New York and she's never met Olivia as she left before she was born etc.
That is why you need absolute clarity and no excuses for your decision. It's never the nephew you adore that gets sneaked in, it's always some cousin or friend's toddler or small child you've barely met who is brought with a stinking cold and a bag full of noisy toys.