Help on dropped 3rd strike

[Rufus]

I'm coaching a 12U rec team playing under USSSA. Our pitcher threw a pitch that the BR swung at and missed for strike 3. The catcher had to underhand the catch but the ball clearly hit his glove first, then bounced into the glove pocket (i.e., didn't hit the ground, then into the glove).

There was no indication of anything from the PU after the catch. I asked for time, the BU granted it, but the PU overruled and said no time granted. I asked my pitcher to proceed, he toed the plate, the batter stepped into the box, then the PU called the batter out.

Once he did that I understood why he didn't grant time or call the out on strike three and was waiting for one of a couple things to happen:

1. BR runs to 1B
2. Our pitcher or catcher tags him
3. BR "abandons" (don't know the right phrase) his right to advance due to dropped 3rd strike

My questions are -

1. The BU saw the pitch go in the glove and not hit the ground (granted, I'm assuming this because if he saw the play the same way as the PU he wouldn't have granted time). I doubt the PU did because the catcher's body blocked him. Is there any signal, other than a lack of signal, to look for in the future so I can instruct my catcher look for it and know he has to tag the BR?
2. Should he just tag him "in case"? I'm trying to avoid an unnecessary throw to 1B, potentially allowing base runners to advance, if possible.
3. Is there any mechanic the BU can use to indicate the ball was caught and a tag isn't needed? In no way, shape, or form would I want to instruct officials before, during, or after the game. I guess I'm trying to figure out if there is supposed to be coordination between the two in that situation.

Thanks in advance.