Team A is winning 54-53 when A-1 is fouled with 1.3 seconds left in the game. A-1 goes to the line to shoot two shoots.
the coach of team B informs the official nearest him that he would like a timeout if the second shot is made. A-1 misses the first shot and the second shot hits the back of the rim and bounces toward midcourt. The official nearest coach of team B blows the whistle to give team B a timeout as B-1 bats the ball out of bounds. There is .4 seconds on the clock when the official blows whistle to give timeout. What should the ruling be?

What we have here is an inadvertant whistle situation. You say that the official blew his whistle "as B1 bats the ball out of bounds," so I presume that this is before the ball has gone out. Go to the AP arrow.

Hope that the timer isn't fast enough to stop the clock before it runs out and declare the game over. :D

If the timer does stop the clock, you have to use the AP arrow for possession at the nearest spot. The team can still have their TO if they wish now.

Oh yeah, the partner that blew that whistle is buying drinks for the night! :)