Interrupted dribble
 

Tonight BV: A1 pushes the ball up the court in transition. He pulls up and loses control. As he chases the ball down, it touches B1's foot. A1 grabs the ball in both hands, then starts another dribble. Violation?

I think you are taking liberties with the well defined term 'interrupted dribble'. A more apropos description for your sitch is 'fumble' followed by a touch by an opponent. I would not have a violation here even if the touch were by a teammate of the former dribbler. The only occasion for a violation here is if B1's contact with the ball were deemed intentional.
Don't ask me why, I'm JUSTACOACH:p:p, but I'll take a stab. See 9-5-3...Pg 55
RULE 9 SECTION 5 ILLEGAL DRIBBLE
A player shall not dribble a second time after his/her first dribble has ended,
unless it is after he/she has lost control because of:
ART.1 . . . A try for field goal.
ART. 2 . . . A touch by an opponent.
ART. 3 . . . A pass or fumble which has then touched, or been touched by,
another player.

Not at all. He lost control of the dribble.

It matters not how we define his obvious loss of control, the intervening touch by another player eliminates the possible call for illegal dribble, or travel for that matter. Only potential violation, as I mentioned above, is kicked ball, if you deem intentional contact w/ B player's foot.
I can't imagine why you come with this pedestrian a query. Is it possible you are invested in some of your legendary over parsing of a mundane sitch?

Here's the deal. This happened tonight. After receiving a pass over the top of a press, A1 lost control of the dribble, retrieved the ball in both hands, and started another dribble. I was trail, quite a way behind the play. I did not actually see the ball touch the foot, but Coach A helped me. "He kicked it!" I'm fairly certain that it was not an actual kick, nor was the coach asking for one, but rather was saying that this gave A1 the right to start another dribble. I don't see that it does. If contact by B1 had caused the loss of control, yes, it ends the dribble. But contact was lost before the contact, (I think) so I see this play as a violation. Had I seen contact with the foot, what would I have done? Can't say. But in hindsight, I think I accidentally made the right call.

Now that you have provided the denoument, which does not make mention of any ball contact by another player, I concur and support your surmise of illegal dribble violation.
However..
If the eye-in-the-sky shows the ball accidentally contacting another player, as declared in the OP, I think you got some 'splaining to do...

Why?

No as per NFHS rule 9-5-3. It's a fumble touched or touched by another player...no matter how much over-thinking you might want to inject into it. :)

Whatever it was, it definitely was not a fumble.

So you're saying that an interrupted dribble that hits a defender's foot doesn't grant the previous dribbler a new dribble, but a fumble that hits such a foot does?

I didn't say that, I asked the question. If an interrupted dribble is touched by an opponent, does that end the dribble? I see no rule which says it does.

You did say that; at least that's how I read this post:

Unless you're also saying a fumble would give you the same ruling as your post above.

I Do ...
 

NFHS 9-5: A player shall not dribble a second time after his/her first dribble has ended, unless it is after he/she has lost control because of:
ART. 1 A try for field goal.
ART. 2 A touch by an opponent.
ART. 3 A pass or fumble which has then touched, or been touched by, another player.

Nobody mentioned a fumble until JR. There definitely was not a fumble. Player had a dribble, then lost control resulting in an interrupted dribble. Ball then touched/was touched by the defender. Dribbler then picked the ball up with both hands. Can he legally start a new dribble after this sequence? I don't see how, but I entertain other opinions.

Who cares?

Read case book play 4.15.4SitE. That says the defensive touch ended the dribble. In that case, what difference does it make what type of dribble it ended... a regular ol' dribble or an interrupted dribble?

Rules rulz!