Finally have some time to get back to this:
Relay U1 is wired to latch on its own contact. When the level in the tank is between full and empty, the NC Full float switch is closed. As the tank drains, eventually, the NO Empty float switch closes, causing U1 to pull-in. As soon as it does, U1's own contact closes, meaning that it doesn't matter what state the Empty float is in. Assuming the valve is operated, the tank begins filling. When the level rises to where it opens the Full Float switch, that depowers U1, and it shuts off.
This is depicted in the simulation by the red and green traces. The Empty float switch is closed while the Red trace (a simulation control signal) is >0V. The Full Float switch is open while the Green trace is >0V.
The Blue trace in the lower plot pane shows the state of the Latch relay U1. Note that it turns on when the Empty switch first closes, and turns off when the Full switch first opens. The second set of contacts on U1 Create a signal that is actively driven to +12V, or to ground.
If you were using a normal solenoid valve, this is all you would need, because the Latch relay could control it directly.
The low-to-high edge of the Latch signal is used to pulse relay U2. Note use of steering diode D1 to create a pulse. The value of C1 and U2's coil resistance controls the time duration of the pulse. D2 steers the high-to-low edge of Latch to relay U3.
The contacts of U2 and U3 create an H-bridge which can drive current through the valve coil L1 in either direction. The current through L1 is shown in the Purple trace. Diodes D4 and D5 are snubber diodes which protect the contacts in U2 and U3.
I would use a single 12V supply to do everything. A 12V short duration pulse wont hurt your valve. I would look for 12V relays that have as high a coil resistance as possible. The cheap 12V Bosch automotive relays typically have a coil resistance of 85Ω, which will require a huge capacitor. I have some relays around here where the coil has polarity marks. These are internally biased with a permanent magnet, and have a coil resistance of ~500Ω. Those are the ones I modelled in the sim.