First, I sure hope the rabbit isn't at your place. At one month old, she should still be with her mother for at least 3 weeks.
Lionhead isn't recognized as a breed in most countries - where I live, there is no such thing as a purebred lionhead. The reason is that lionheads are just the product of angoras mixed with regular rabbits. The babies who had more or less a 'mane' were deemed 'lionheads' and reproduced together, but it is not at all stable : you can't predict if the kits will have long hair / short hair / some kind of mane, or how the 'mane' will turn out. When you look at them, you can find a lot of lionheads who have a real fluffy mane as babies and turn out to have just a few weird long strands as adults (it's what happens to most petshop lionheads, who generally don't look that good as grown ups, at least for those I saw). I once had a rabbit bred from a 'lionhead', he was kind of a lop with a kinda fluffy fur and some of his siblings had more or less of a mane but not all of them. Your bunny is still very young, but she doesn't look very fluffy to me. Actually, she looks just like the rabbit I had did around the same age - so I might be wrong, but I'd say you've got a short-haired rabbit who might have some thicker fur than most as an adult...