That first circuit is useless for this task. The valve requires an analog value between 0 and 10V.
Put an RC smoothing (lowpass) filter direct from the Arduino. Use at least 10k to swamp the difference in the Arduino output's source and sink impedance. That should give you 0 to 5V analog voltage from the Arduino's 0 to 100% PWM signal.
Put that in an Op Amp with a gain of 2, with a V+ of 12V and a V- of -2V or more.