You seem fairly certain that it's constantly broadcasting. I'm curious what your source is? I have yet to find anywhere that provides a definitive answer as to which method of active RFID is used by MBs.
I'm not saying a MB isn't constantly broadcasting - it may very well be, but active RFID does not have to be constantly broadcasting, so it's entirely possible that the active RFID tag is only broadcasting its ID when prompted by a signal from a receiver. In this case, the transceiver operates in low-power mode until it gets a request from a receiver, then wakes up, broadcasts the ID, and then goes back in to low-power mode. It still drains the battery, but at a much slower rate than constant broadcasting - even more so outside of the parks since it's not being asked to broadcast its ID.