thrown bat injuring a catcher

Question
I was the plate umpire in a rookie ball playoff game.  Tie ballgame, runner at second, none out.  The batter hits the ball, then throws his bat, dropping the catcher, caught him in the ribs.  Runner from second makes third and is looking at home.  I call dead ball, called the batter out at first base and return the runner to second.  The offensive team says the rule is a warning is to be given for the first thrown bat and the batter is only called out on a second offense, the fact that the bat hit the catcher is irrelevant.  I stood my ground, he protested.  The ruling came back that I was supposed to wait till play stops with an injured player on the field.  No out was to be called, offense has runners at first and third and just a warning issued for the thrown bat.  The injured catcher was pulled from the game.  There are rules to cover thrown bats and injured players, but, nothing about a thrown bat injuring a player on the same play.  Got anything on this?  I still think I was right for calling him out.

Answer
This is a difficult question since there are many different interpretations depending on level of play. I will assume this was not milb rookie league playoffs that are happening now here in Florida.
First, definitely not a dead ball unless the bat interfered with the catcher, let the play finish. Many youth leagues have a warning on first offense. If it was intentional, its an ejection, again, not a dead ball, let the play continue. Lastly, its is not an just an out for repeat offenders, it is an ejection.
Is it possible that you may have used personal feelings or judgement on this one play? Again, this is a difficult question to answer. Consult the local league rules. And remember, unless its interference, let the play finish.