And how to get an invitation? God, sorry, so many questions, haha. Yes, in the guild they say you need an invitation. And excuse my English))
I went back and played through that part of the game. You get the Invitation from Lumi when reporting that you purified the West Theospolis Spirit Nest. You quite literally can't miss getting it. The only catch is that it doesn't show up as an item in your inventory until
after you've attended the Mayor's Ball, so you can't see that you've received it beforehand.
To actually get to the Ball, you need to buy the Formal Suit from Thayer in Miriko Village. Fynn pays for it, so you don't need the 4,000g it's priced at. Afterwards, the game sends you directly to the Ball via cutscene.
If you can't get the Formal Suit from Thayer, then you definitely did not report purifying the West Theospolis Spirit Nest to Lumi. You can double check by going into your inventory and scrolling all the way to the bottom of the Seduce Items tab. When you get the Eternal Spring Water from West Theospolis after purifying the Spirit Nest, it shows up as the last item in your inventory, and after you report to Lumi it'll be marked as Complete.