Well, WBT follows a similar line of thinking...we don't want our kids to raise their hands to answer questions during a lesson, because, you're right-- they disengage. We want to call on any kid at any time, and make our classroom a place where kids can celebrate when they can demonstrate they understand ("Woo!") and where they know it's ok if they don't fully understand the first time ("It's cool!"). However, if a child in your room wants to speak during a time where they are not supposed to be speaking freely, how do they indicate this to you?
If there is no signal for the kids to give to you when they want to speak during a time that may otherwise be inappropriate, then that's probably an issue that the teachers and administration may need to address, as I'm guessing it leaves teachers with no real way to help kids control themselves when they have something to say. Think about it, if you're a kid, and there is no signal to tell the teacher you want to speak, you'd probably just blurt it out, because what else can you do to make sure you are heard? I would assume that, without a silent signal of "I have something to say / ask", you have kids speaking out of turn and calling out all over the place.
Now-- since 'hand-raising' has a pretty taboo connotation in your school, give them a different signal, such as making a fist and holding it in front of them, or something like that. Then, your Rule 1 can be "Make a fist for permission to speak." or whatever you choose.... maybe making a "thumbs-up" sign that they touch to their shoulder...any signal would work. Hand-raising is nice because it's so obviously sticking out of the crowd, but you can find a different signal that works. Whatever it is, students should have some way to let you know they'd like to speak, have a question, or need to request something when it's during a time that kids are asked not speak freely.
And...as far as kids disengaging when they feel like it, well, that's what all this is for! Your WBT journey begins a lovely time in your classroom where the kids can't HELP but be engaged, because you leave them no time to disengage (and most end up not wanting to anyway, because they're having so much silly fun.)
This is a tough situation for you to be in, but I'm sure if you talked with your administration and explained that you'd like your students to be able to give you an orderly signal that they have something to say, that you'd be supported. In the end, if they don't support any signal indicating a desire to speak or ask a question, then you'll have to throw Rule 1 out...but I bet it won't last forever. I don't see how they can't see that it's necessary for a classroom to remain organized and efficient, optimized for learning.