The thing with the Faceless Men is that they are VERY expensive. It's said in the books that the costs of hiring one faceless man is equal to the amount it would take to hire an army.
Tywin would have to sacrifice a huge part of his wealth to kill one person. The Faceless Men are the elite assassins of Planetos and to hire one of them, one has to part with the greatest part of what they find valuable and most people aren't willing to do that. It's easier to hire cheaper, less talented assassins.
Regarding Tywin......I forget about how it went down in the show, but in the books......Tywin didn't really believed the hype about the Dragon Queen beyond the Narrow Sea. Most people simply think she is a rumor, spread by drunken sailors.