Is there any known case where height neurosis was successfully cured with therapy or something like that?
I mean if it's curable that way, then it would be a cheaper solution than LL.
It depends. If your height is 190, your girlfriend is a hot model, you live in a big house and the only reason for your height neurosis is that your sibling bullied you when you were younger then chances are it can be cured. In other words, whenever your perception of yourself is not aligned with how the world views you, then talking to someone in said world may help.
However, most people with height neurosis suffer from their condition because they are at a disadvantage in social surroundings, in the dating market and at work. Some people begin to avoid all forms of real social contact, which in turn has a detrimental effect on their social skills, makes them more anxious and less creative. This in turn decreases their chances on the dating market, at work and in social surroundings even further. It's a vicious cycle, but it never started with the height in the first place. It always starts with the realisation that we cannot have something we desire on account of our height.
If a psychiatrist could turn a short guy into a confident Tinder god then this might cure the neurosis but I am fairly sure no psychiatrist will succeed at this.