My sub was not allowed to enter and a buzzer in the middle of a FT attempt

[CoachJW]

I had a couple of new situations last week and they happened in the span of 20 seconds.

A1 is shooting one free throw. A6 is at the table to report for the shooter. Team A is on the road at a league opponent's gym, early fourth quarter of a close game. The official nearest the table beckons A6 in to the game, is informed that the sub is for the shooter. The other official begins to administer the free throw. The ball is in A1's hands, and during his free throw attempt the table buzzes to alert the official that there's a sub at the table. The free throw attempt misses.

Right away, as the visiting coach, I'm arguing that we get to shoot the free throw again. If that's not in the rules, I don't see what is keeping anybody at any scorer's table from continually messing up an opponent's free throw by buzzing the horn. In the moment I didn't know of a rule that I could stand on, but it didn't seem right.

Play is continuing. We rebound, miss, rebound again, and make a shot (OK, I've dropped the first argument because now we've got 2 points instead of 1...). After the shot goes through we get a whistle and one of the officials needs to check something at the scorer's table, a discrepancy in the score or something. One official beckons my sub in the game, while the other denies the player from going in because the dead ball was not a "natural" dead ball, but rather an official's timeout. I had not heard of that rule, couldn't find it with a quick look at my rule book, so I thought I'd post it here. Thanks for your help.

[ref2coach]

Not allowing the sub was an error by the official.

[just another ref]

Not awarding another free throw was not an error.

[Loudwhistle]

If the clock is stopped and the ball is dead then subs can come in. Even if it is an official's time out. in your case the official was wrong.