We used gate hooks, high up on the door for our twins. They have a bit of give , so not a risk if there is a fire. And yes I know people will not agree with it, but you have to have lived with this to understand just what it is like. They didn't need a lot of sleep and had a habit of wandering the house, found them in the kitchen on worktop smashing all my glasses to the floor,was just one thing the did, the list is huge. Like your son they just climbed the stairgates. I know how hard it must be for you, it really is tiring and wears you down so much, It will get better, trying to stay one step ahead of them is so hard and the baby books are really no help. You are doing everything you can.
Good luck, you sound like a caring mum who is at the end of her tether, is there anyone who can give you a break?