Pinch Hitting Problems

I became involved with a unique situation tonight in our District level Juniors Little League All Star Tournament game and would like any umpires or coaches out there to chime in on the ruling:

With one out in the 4th inning, I called time and alerted the umpire that I wished to announce a substitution of a legal player on the bench as a pinch hitter in place of a starter. The umpire refused to allow the substitution stating that all substitutions needed to be made between innings. This seems to go directly against the substitution rules for pinch hitters as stated in Rules 3.06, 3.07, and 3.08, which does not allow for the umpire to reject a legal substitution.

I was sorely tempted to protest, but deferred, not wanting to incur the wrath of an umpire that has a reputation for vindictiveness. But in 10 years of coaching, I have never seen the substitution rule interpreted like that.

So, very simply, does Little League rules allow for a pinch hitter in the middle of an inning?

Any experience with this situation, one way or the other would be greatly appreciated.