How can one deal with a person who commits major sins and lives in the same house, given the non-application of Sharia and the impossibility of leaving the house or confronting the perpetrator?

The way to deal with a transgressor is, first, to advise him gently and wisely, for advising a Muslim brother and rebuking him for sins is a religious obligation. This advice is especially emphasized when witnessing him committing an evil deed. If he persists in committing the evil deed, then leave his presence and do not sit with him. If you predominantly believe he will not respond, then limit your interaction with him to what is absolutely necessary, and abandon him with a gracious abandonment.