I don't know how much extra weight is the minimum that your character's superpower will have to simulate to enable them to break through a brick wall at a normal walking pace.
But I can suggest that a simulated weight in the weight range of elephants should be sufficient to break through a quite strong wall, even if not necessarily to walk through the wall.
I once found a century or so old article reprinted on the internet where a circus man wrote about elephants. He mentioned a fight between two bull elephants in the elephant barn of the circus's winter quarters. They fought until one elephant's tusk broke off and the other one pushed him through the brick wall, 14 inches thick, of the elephant barn. Despite the vast amounts of blood they shed, both elephants had only minor injuries except for the broken tusk.
And in another article I read about a contemporary villager in India who was injured by a flying brick when a hungry elephant broke a hole in the wall of his house and reached in to grab a bag of rice.
And in another century old article I read about a circus parade in an American town where the elephants were stampeded by a barking dog and ran through the front wall of a wooden house, through the house, and out through the back wall.
So I looked up more such stories just now.
Here is a link to a video of elephants passing through a gap in a concrete wall.
Here is a link to a video of elephants destroying a wall:
p. 73 here:
And if you don't think that mere elephants are heavy enough to simply walk through strong walls at a normal pace, just give your character simulated weight that is twice a much as an elephant, or 5 times as much, or 10 times as much, whatever seems to be sufficient.