I'm in a similar position regarding the height and age, so here is another perspective for you to look at it.
Our height is in the range of super short, not just short. Just speaking from a scientic point of view, people do have a subconscious impression of how capable a short guy is, no matter how confident you are. Unless you take your time for every person you meet, so that person finds out you are cool with your current height, they probably won't assume it before knowing you.
The short height is only a passive disadvantage, but no deal breaker for living to the fullest potential. When you actively show your personality, you can prove people wrong.
But there are so many circumstances where you are judged and opportunities slip by before you even have the change to show that you are unaffected by your height. And this is all passive. I guess this isn't something new to you.
So it is just fair and it makes sense to put oneself back in a neutral position.
Life is too fun and short to refrain from doing what you want because of what others think of you. This thought should not even cross the mind of people. This thinking pattern is cancer for a fulfilling life, being short ot not.
For my LL, no one will have a word of interference. They will be informed when it's already in the process or casually some other day, just so they know.
Be totally selfish with your wish to do LL. It's to widen your window of opportunities and options, and to be able to do social stuff casually. Currently, even just talking to people on the go seems try hard just because it really is - from a physical point of view. So yeah, do LL for yourself, and use these explanations (not justifications) why it just made sense to you to do LL.
And about the costs... People buy a new car or put themselves in debt with that money and start paying off a house that immobilzes them forever. And both of these objects are liabilities that drain money. You invest in yourself. This is the smartest investment people can make. Some invest even more money in further education from seminars and books that help them to become independent. You're on the right track.
I always say, you are 100% responsible for what you want to get in life, no matter your circumstances.
At the same time you a free to do what you want, without having to justify or explain why. If you feel like it, that is enough of a reason.