That's nice, also I will have one more bit of accuracy.
Well, not really. (Basically, it is lost in the common mode values, which comprise half of all valid value pairs.)
Hey, I think I have a better option.
Use a single MOSFET to switch the 5V line based on the optoisolator output; configured so that at startup (and when the microcontroller side is not powered but this side is), we have a 0V line, and 5V only when we get the enable signal from the microcontroller. Let's say there is a ferrite bead and a capacitor to filter this MOSFET-controlled 5V line, and call it the reference voltage.
(One could use another LM358D opamp pair to buffer the 5V reference line, but I don't know if that is necessary; I too am just a hobbyist on the electronics side.)
This reference voltage line is used for the VREF, and for the noninverting inputs of the two opamps.
The AD5663 inputs are connected to the inverting inputs of the opamps, with the opamps in differential amplifier configuration. Each output can handle a load of 2kOhm (or larger), so let's make the resistor between the AD5663 output and the inverting input of the opamp at least that. To get amplification by factor 4, the feedback resistor for the opamp needs to be four times that. Let's say they are 2.5kOhm and 10kOhm (0.1%). On the noninverting side, we need a voltage divider where the resistor between MOSFET-controlled 5V line and the noninverting input is exactly one and a half times the value of the resistor between the noninverting input and ground; for example, 5.0kOhm and 7.5kOhm. This way it will be doubled.
With this, when the reference voltage is low (0V), both outputs are 0V too.
When the reference voltage is 5V, the output swings from +10V to -10V. DAC code 0x0000=+10V, 0x4000=+5V, 0x8000=0V, 0xC000=-5V, 0xFFFF=-10V.
You could even use only +12V, GND, and -12V, by using a voltage reference between +12V and GND for the 5V reference voltage, perhaps buffered by an opamp per noninverting input to a difference amplifier (and the VREF to AD5663 directly from the shut); but with the MOSFET as a switch between the reference and +12V (possibly with a transistor to invert the gate logic if using a P-channel MOSFET) basically turning off the reference unless ENABLE signal is present. This would give better precision, I think.