Your English is pretty good. It is clear. You've written a long, complex sentence. It is grammatical. A native speaker would make different choices of words and word ordering.
"Tickling" is not a good word choice. "Tickling" literally means this:
https://www.youtube.com/watch?v=8W5In5gyiII
"Tickling" feels good and bad at the same time. It means something that makes you laugh.
"Nagging" or "bothering" would be better.
With some other small corrections, you could say:
"Here is a question which is nagging at me: at school, why were we told to write "have got" instead of "have" in sentences when, only now, I learn that writing "have got" is colloquial?"
The answer to your question is: schools are like that!