The ultimate objective would then be to get the signal through 2 AD630s and harvest the 2 distinct DCs

The problem you have is that you won't necessarily have phase coherence between your arduino output square wave and your reference frequency so, as the two drift apart then back together (almost randomly sometimes I expect) you will get a vast variation in the output of each AD630 multiplier.

Some folk solve this by using two multipliers for a single channel and adding the two output signals like so: -

X = \$\sqrt{A^2+B^2}\$

Now this would produce a constant output signal if your frequencies matched but the phases were not locked-in.

The ultimate objective would then be to get the signal through 2 AD630s and harvest the 2 distinct DCs

The problem you have is that you won't necessarily have phase coherence between your arduino output square wave and your reference frequency so, as the two drift apart then back together (almost randomly sometimes I expect) you will get a vast variation in the output of each AD630 multiplier.

Some folk solve this by using two multipliers for a single channel and adding the two output signals like so: -

X = \$\sqrt{A^2+B^2}\$

Now this would produce a constant output signal if your frequencies matched but the phases were not locked-in.

My understanding is that AD630 prefers AC coupled signals for both reference and the actual input

No, that is incorrect. You must use DC coupling or a pull-down resistor to 0 volts after the input capacitor to leach away the input bias currents.