I had something similar happen in a game this last season. Seconds left in the 4th quarter with a tie game. Team A dribbler drives the lane, goes airborne for a shot, gets fouled and releases the shot. I hit my whistle for the foul and the horn sounds a fraction of a second later. The shot does not go in, so we shoot two with the lane cleared (he missed them both).

So... you are going to allow B1 to linger before you administrate the Free Throw to A1. You are playing right into his hands. You are helping B1 'Ice' the shooter.
We got grilled from one of our observers because we did not administrate the Free Throw in an appropriate time frame. Once I informed B1 that if he did not move, then A1 was going to shoot the free throw until he made it. That got him to obey.
Unless the game clock was staring us in the face we were not going to be able to observe the time. Our lag time to turn our heads to see the clock would have been slower than the lag time of the game clock timekeeper.

Correct as per NFHS rule 5-6-2 EXCEPTION3- "If a foul occurs so near the expiration of time that the timer cannot get the clock stopped BEFORE time expires or after time expires, but while the ball is in flight for a try or tap for field goal, the quarter or extra period ends when the free throw(s) and all related activity gave been completed."

NCAA rule 5-7-3(c) basically says the exact same thing.

I am going to "ask" him to go where he belongs. If he refuses, I will proceed with the delay process. You chose a different way to get through this and I applaud your efforts. However, I don't believe the rules support administering a free throw with anyone in the lane when the ball is to become dead after the final shot. If the rules & actions support my call, I never have to worry about my decision.

With the clock, I would probably get rid of the lagtime term. This term was removed from the NFHS manual a couple of years ago. On every play, there is a 3rd official away who should be all over the clock. Things can get complicated if you are in an auxiliary gym with only one clock on the baseline. The way I read the OP was that the whistle clearly sounded before the expiration of time. If that is the case, the crew needs to be really good and know the remaining time. Especially true at the end of a close ballgame.

Just curious, so a few questions:

Would you hold up the administration of a normal FT if a player were standing outside of the 3pt line, but well below the FT line extended?

Would you ask this player to go where he belongs?

What penalty would you assess if he doesn't move?

Yes, Yes, T.

What do you do if you notice A and B occupying the "opposite spaces" before you administer a FT? What if B chooses not to occupy a space and A puts an "extra" player in there? What if B tries to go "below" the block?

I ask them to move, and they do. If they refused, then I'd deal with that.

Bob,
I agree that the officials should check that the players are properly lined up prior to administering the FT.

I'm not sure that a T is the proper penalty. What rule would you point to for issuing a T here?

What's "the delay process?"
While lagtime isn't in the book; all that means is we can put .5 back up if that's what we see. The problem is, there is still a human reaction time that is going to take a brief moment to look at the clock. As has been pointed out, a lot of times you'll have foul, horn, whistle.
And in the situations where you have foul, whistle (at, say .3 seconds), horn, I think you're expecting too much of the off-call officials to react quickly enough to see any time on the clock.

Especially if you're working two person with only one scoreboard in the gym.

True, but even with three of you and 5 clocks, with that little time on the clock, it's easy to miss. Especially on, say, a shot at the FT line.

Failing to allow the ball to become live -- 10-3-6(?) The same rule used when the FTer is not in the semi-circle.

Fwiw....+1

Just tell the player what the rule is. If the player doesn't want to follow a rule, it's an easy "T" under 10-3-6. And if the l'il sh!t is still adamant about not moving and his coach doesn't want to do anything about it either, I ain't gonna argue with him/them any further. Give 'em a minute to comply and then it's Hasta La Vista; forfeit under 5-4-1. Put the onus where it belongs- on the person(s) causing the problem. And that includes the head coach who is supposed to be in control of his players.

We should never have to plead with anybody to have a simple rule followed.

I am fairly convinced that if this situation took place, I could say what I needed to say to not have to give a T at least in a situation where the ball is dead. I think this is a very good question, but sometimes we have to figure out how to talk to people and get what we want, even with some conflict as apart of the situation.

Peace

Again, I'm an official in other sports, but not basketball. I have to ask, though, how this is possible (assuming we're not in a running clock game).

Either the foul happened before the buzzer, thus leaving SOME time on the clock... or it didn't, thus there was no foul.

Human reaction time, to be blunt. :)

There's the reaction time of the person running the clock. Then there's the reaction time of the non-calling officials hearing the whistle then looking at the clock.

Even if we know there was a short time lag, if we don't see how much time to put back on the clock, by rule we can't guess.

Also, as has been pointed out, 99% of the time, there will be a lag between the actual foul and the whistle. The contact/foul could happen with a second remaining. The whistle could blow with .2 second remaining. The timer's reaction time takes .2 second, and so does the off-ball official's.

In NFHS, we don't get to use video replays for this.