Thank you!
I have just done the process as described by you and the fellow posters, and so far the clicking is fixed.
I did it like 5 days ago and drove around 400km so far, and the clicking hasn't returned. Likewise, I will keep you guys updated if it comes back in the future.
The process itself was pretty fast (around 20 minutes), mostly because I couldn't fit my hand really well in the gap even with the direction lowered at maximum, but it's manageable. One thing to keep in mind is to be careful of not dropping the flat key inside, I imagine it would be pretty hard to get it back.