My thoughts:
The writer says that her partner "tried" to give his son a timeout for mishandling the rabbit, which "just resulted in a lot of very loud screaming," suggesting the timeout wasn't actually enforced. I am wondering if there is an overall discipline issue? I can understand a small child getting overexcited and not realizing their own strength, or forgetting some nuances of animal handling, but I guess I don't know why someone had to write to an advice columnist to learn, "Keep the rabbit and child physically separated until the child is more responsible." Isn't that the next logical step? It makes me wonder what other basic things the writer and her partner haven't thought of, and there's a lot the letter doesn't explain.
For example, it's implied the writer, partner, child, and rabbit all live together. Is this a new situation? In what order did beings move in? Like, did the writer have this pet rabbit and then decided to move in with her partner and child (or vice versa), or were the three humans living together and one decided to get a pet rabbit? Before bringing a delicate pet and small child together I would think long and hard about how to avoid injury to both--it's not exactly inconceivable that a small child might get overexcited by a small animal. If from the start they had been able to stay, "This is the rabbit room. You will not enter the rabbit room alone," that would be a lot clearer to a kid than nuances about being gentle. And you could put a lock on the door easily enough.
The writer describes the child as "chasing the rabbit around the house," which implies more of a free-range situation, like you would have with a dog or a cat. I don't know why anyone would think that was a good idea in a house where a small child was also free-range (cats and dogs are usually better at escaping inside a house than rabbits are, though I admit I don't know much about rabbits). If the child was too rough with the cat or dog, yeah, you'd have to find a way to physically separate them, but that would also take away a big part of the enjoyment that people get from their cats and dogs, to just have them in a room or large cage all the time except when you could actively supervise. The writer also says they "found him" holding the rabbit by the ears (again!) like both adults were in a different room from the kid and rabbit, which suggests that proper supervision is NOT happening in the first place.
So, the vibe I kind of get is that the adults didn't think about the consequences of introducing a delicate pet and a small child to the same space, haven't been supervising the two together properly, don't know how to discipline the child effectively, are shocked that the child can't follow directions, and have now completely run out of ideas. Maybe she SHOULD give that rabbit away, I'm beginning to think she's not a good caregiver no matter what...