The Aussie batsman, who has been captaining Middlesex in County, was forced to miss a match on Sunday after being forced into isolation following a positive test result.
Several former players, including wicketkeeper-batsman Ian Healy and England's Kevin Pietersen, were critical of Paine's captaincy.
The 29-year-old joins the squad to fill a void left by the retirement of George Bailey at the end of last year's BBL edition.
All-rounder Glenn Maxwell, who hasn't played for Australia since last October, made the cut to the list, while Handscomb, Shaun Marsh and Nathan Coulter-Nile were ignored.
Australian cricketer Peter Handscomb has said that while it's disappointing to not be in England for the County season, there is a bigger picture at play.
Peter Hanscomb has so far made 40 appearances for Australia in all three formats of the game and is the current captain of Victoria and won three titles with them.