The Senate can refuse to seat him, highly unlikely. South Carolina cannot recall him. He was reelected in 2014, so primarying him can’t happen until 2020. Not too many options beyond forcing him to resign.
There must be plenty of dirt out there, and I want to lead this... senator.... into investigations where it will be my uh, dearest desire that he will be led into resignation.
So the little worm continues to flourish in the swamp! Sad.