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.
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.