I have always thought that when using the bridge, you needed to keep a hand on the bridge. Today, in a league tournament, someone took their hand off the bridge and shot. There was a discussion and tournament director had the game be played over as they could find no rule about it. This is BCA 8 ball.
Is it a foul to take your hand off the bridge when shooting with it?
I think you want to look at rule 1-3 "Use of Equipment" in the BCAPL rules. Then you have to make an interpretation of whether the wording "You may not use equipment or accessory items in a manner other than their intended use" applies to not holding the bridge handle. Then you have to decide what the penalty is because a standard for violation of 1-3 is not stated.
It was a mistake to replay the game. A decision should have been made at the time of the shot. I would have said that the player got no unfair benefit from what some (but not all) might possibly consider an unusual way to use the bridge. The same method is available to everyone who wants to use it.
In the BCA/WPA rules the only rule that might be applied is under Unsportsmanlike Conduct (6.17(h)) "using equipment inappropriately" which is a little nebulous. If the ref/TD feels that letting go of the bridge handle is not an appropriate way to use the bridge, then he should apply an appropriate penalty. I think the use is appropriate, but if not, then a simple warning seems like the right penalty.
If you watch snooker, you will see the following use of the bridge:
You need to get the cue up higher than usual to bridge over a ball and both balls are far from you. (Cue ball on the other side of the blue ball on a snooker table.) Grip the bridge very firmly with your left fist about a foot from the butt end. Place the knuckles of your fist on the table and twist so that the butt touches the cloth and the bridge head is elevated off the cloth. Such a shot was played in the current World Championships and I have seen Ronnie O'Sullivan use the technique. In my view such use is appropriate and the shooter gets no unfair benefit from using the bridge like that.