Electronic – Bistable solenoid unlatching voltage

This isn't too hard to implement. I can see the box and controller getting out of sync, but if the controller hits the zone twice and it doesn't matter what order the two sub-zones come on in, then that seems fine.

You are right in that you need some kind of memory. Since the unit will have no power to it between uses, that memory needs to be non-volatile. A microcontroller with built in EEPROM would do fine. EEPROMs are only good for a finite number of writes, but that's 100s of 1000s at least so no issue there.

When the power to the switch box turns on, all it really does is run the micro. The micro then turns on one of N relays to route the power to one of the sub-zones. It also writes the new state to its EEPROM so that it will power the next sub-zone in sequence next time.

A tiny micro running at slow clock speed can easily handle this. The 5V current will be small so a linear regulator will do well enough and be simple. Get relays that can run directly from the full wave rectified AC so there are no power conversion issues. 24V AC after full wave bridge with filter cap should be around 30-32 Volts. "24V" DC relays would work but get a little warm. Genuine 30V relays may be harder to find, so you could get 24V relays and put a resistor in series with the coil. A reverse catch diode accross the coil and a NPN transistor with base resistor to the micro is all you need per sub-zone output.

Another thing to consider is that the micro needs to see one power up each time the main controller turns on the zone. This should be as simple as putting a little low pass filtering on the micro's reset input so that it doesn't start running until a 100 ms or so after power is applied. By that time glitches and switching transients should be over.

The main controller also needs to leave some off time between powering this zone so that it toggles to the next sub-zone. It will take some time for the voltage to drop before the micro loses power or is shut down by the reset circuit. It could be a second or two depending on what values are chosen.

The more I think about it, the more I'm realizing the trickiest part of this is the reset circuit. You want to make sure the micro runs cleanly once per power up, and that it goes into reset cleanly once on power down and not too long after power down. This is all quite doable, but something that needs to be considered.

This isn't a direct answer but is pointing out a alternative.

You don't necessarily need a double ended drive like a H bridge. That would certainly work, but a push/pull driver with a capacitor in series with the relay coil would do it too. Here is a possibility to think about:

In the steady state, the capacitor charges up to whatever the drive level is at the left side of the relay coil. It then provides the opposite polarity for the right end for a while immediately after the left end is switched. 22 µF will charge up to only 1.4V after 300 mA for 100 µs. 22 µF and 16 V can be had from a 1206 ceramic capacitor, like this one from Mouser. The cap can be polarized since the top side will always be at or above the bottom side.

The double emitter follower can be driven directly by a CMOS logic gate. There are some that can handle this voltage. The input to that could be driven by a open collector with pullup. Since the CMOS logic input is high impedance and you don't need really fast switching, the pullup can be quite high. 100 kΩ should be low enough to work well, but high enough that the quiescent current in the low state is small.

Of course you could replace the transistors with a half bridge drive chip that takes logical level input for higher integration, but also likely higher cost.

Added:

You are asking about driving this from a single open drain output. As I said above, I'd use a CMOS gate that can run from 15V. The high impedance of the CMOS gate input allows for a high value pullup resistor, as I mentioned previously. Here is this concept shown explicitly:

Q3 is a switch. When off, R1 pulls the input of IC1A high for one relay state. When Q3 is on, the input to IC1A will be low for the other relay state. Q3 could instead be the output transistor in the driver chip you mentioned. However, it only takes one NPN and one resistor to replace each channel of that chip. The left side of R2 can be directly driven by your microcontroller output. The driver chip could be less board space, but the NPN and resistor will be cheaper. The whole circuit from the micro up to C1 could be replaced by a half bridge driver chip, which again will be more cost but maybe less board space. Everything is a tradeoff.

I also flipped the relay coil and C1. Since these are in series, it doesn't matter to the operation of the circuit. However, it may be convenient to tie one end of all the relay coils to ground. This second circuit allows you to do that.