Is my inversion correct and is it the one used in an informal letter written to a friend?
"Although satisfied with facilities I had in my previous place, I had a bully neighbour driving me crazy. As a result of this I preferred to move out of my place."
The "although satisfied" seems odd here; you might say "While/Although I was satisfied with the facilities I had...
Also note that the word order "a bully neighbor" suggests that "bully" is being used as an adjective. There is a very old-fashioned adjective "bully" (which was much favored by, and is strongly associeated with, Theodore Roosevelt) which means "excellent; superior; first-rate". I doubt this is what you mean; if you mean your neighbor was aggressive and intimidating, you should say "I had a neighbor who was a bully and who was driving me crazy."
"Although satisfied with facilities I had in my previous place, I had a neighbour who was a bully. As a result of this I preferred to move out of my place."
Based on the cited grammar, could you please let me know whether my reduction is correct or not?
Here are the main mistakes - a missing "the" and a preposition that I don't like, as well as a case of severe wordiness at the end: "Although satisfied with the facilities I had at my previous place, I had a neighbour who was a bully, and so I preferred to move out."