If it’s foul and he’s not certain he can throw the runner out, he should let it drop to keep the runner on 3rd and take your chances on continuing the at bat. If it’s fair, he must catch it and do his best to make a strong accurate throw to the plate. If the ball is not too deep, the runner will most likely tag and test the arm. The 3rd base coach, who has now moved at least halfway to the plate will tell the runner if it’s necessary to go back based upon the direction, strength, and elevation of the throw.
In order for the batter to be out from making contact with a ball while “out of the box”, the batter needs to have one or both feet on the ground and entirely out of the box. If the chalk is eroded, you will never get the call, so don’t bother asking for it. Where you will get the call from the umpire is on RHH push and LHH drag bunts because if the RHH’s foot is in front of the plate, it can’t be in the box and the line of intersection for the box and the 1st base foul line are rarely eroded so the umpire can clearly see it–if he’s looking for it. If the batter’s front foot is in the air out of the box, he is considered to be in a legal stance within the box. If the ball, in the example above, is fair which means it has either bounded over the bag or has first touched ground in fair territory beyond the bag, then play continues. If it does not touch the ground before the bag and first makes contact with the ground entirely in foul territory beyond the bag, then it’s a foul ball.
If you want the bat checked, it should be done as soon as it’s noticed before or during the at bat. Finding out after the fact will not unwind any result. If the batter gets on base, all runs and advancements stand and the bat is removed from the game until it is made to conform with the 18" rule. I would not pitch to a hitter where the pine tar clearly exceeds the limit. As an umpire, I would lay the bat down across home plate and if the tar extends more than an inch beyond that, I’d have the hitter get another bat.
So we know who wins the game This could be a lengthy answer. In the absence of an official scorekeeper, the home book is the official book when there is not agreement between the home and away books. If you are the visitor, it would be smart to check in after each inning and agree upon the current score and pitch counts (if they need to be tracked). The umpire will settle all disputes over score by reading from the home team’s scorebook.