Batting Out of Order?
 

Visitors have a big inning. Next inning, visitors come up again, with B4 batting first. She walks. Home coach immediately comes out with the "official" home scorebook showing that B2 made the last out in the preceding inning, and that B4 was BOO. Visitor scorebook shows B3 making that last out. PU does not know who made the out. How does PU determine if a BOO has occurred?

For me... I'd first make sure both score books "foot", i.e. runs + outs + LOB = number of batters. If so and if it is just a thing like expanding on your example, visitors has B2 stranded at 1B and B3 with F8, but home has B2 with an F8, then I'd try other things to determine whether it was B2 or B3 that ended the inning. My first inclination would be to ask B2 or B3. I trust them more than coaches.

I don't get into scorebooks. If I have a dispute, I tell both scorekeepers to get together and get it settled. But whatever - the home book is official in NFHS contests and that is I will follow. In other contests, the official book should be known before the game.

WMB

I generally agree, no scoring by umpires. However, sometimes an issue like this can be helped by a neutral objective party who sees lineups every day.