I don't follow your schematic. If the relay is a twin coil type why are you driving both coils in parallel? Shouldn't one coil be used to set the relay and the other coil be used to reset it?
Edit:
According to the datasheet A3 goes to +V, A2 is switched to -V to set the relay, A1 is switched to -V to reset it.
Like this
As for whether the FET linked to is suitable, that depends on the gate drive voltage and the relay coil voltage and current. The datasheet shows various coil voltages according to model.