The datasheet does not give the DC coil resistance, but we measured it at 30R.
The datasheet does not give the coil inductance, but we put a step input of 12V into it, and measured the rise of current from this test, we calculated that the coil inductance was 43mH....but we are  unsure if  this test is bogus?.......because doesn't the inductance of solenoid valve coils vary widely throughout the actuation process?
Power source
The power source that we have for valve actuation is provided  by using  power from our 5V rail which we already have in our product.
(i.e. we have a 5V rail present which is provided from a  5W Buck converter.)
Our SEPIC charger uses this 5V as its input)
The answer seems to be on the order of 10,000 uF. But standby for others to chime in.
Whatever the exact final answer is, you should probably double it (at least) to ensure reliability. Or take the opposite approach and calculate the maximum value you can tolerate for recharging in the one second gap.
PS: I notice that solenoid #2 is connected in a bridge arrangement as would be expected, but solenoid #1 isn't. It almost looks like "a typo" in the schematic, on the right hand terminal of solenoid #1.

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