Rovers have been forced to wait while the Grenada international striker acquired the documentation that allows him to play in the UK for the first time.
Bain should finally begin training with Tranmere this week ahead of Saturday's trip to Carlisle.
Barnes said: "Kithson is a big, powerful, fast player who I was impressed with during my time as Jamaica manager and will add some strength to the squad.
"Kithson has scored a lot of goals for the Grenada national team. He was top scorer in the Caribbean Cup and played in the final against Jamaica.
"He was a big reason why Grenada qualified for the Gold Cup, which is a very big achievement."