Well ... I'm sure you are doing something like this:
The aq basic soup is placed in your dish and the naphtha on top so you are at the final extraction step.
I also have this "issue", but IMO NO WORRIES!
The thing is:
Naphtha can hold an insanely higher amount of Spice than you will get by performing your extraction.
Normally you would want to minimize concentration gradients in the aqueous phase and the organic phase. Having an even concentration profile in your aqueous phase would grant that you would have always a high amount of space towards the phase border between both and a higher transition of spice towards the Naphtha.
If you would NOT stirr at all, the concentration would somehow be a little higher on the bottom of your AQ and lower at the border of both phases.
So as you stirr and the WATER phase is moved, there should not be any concentration gradient inside your AQ phase.
But what for the organic phase: It MAY have a severe concentration gradient from the phase border to the top (sorry, I dont really know the english correct terms). Normally you would think that this would decrease the uptake of Spice into the Naphtha more than neccessary.
But as I told in the 1st few sentences the maximum capacity of Spice inside the Naphtha is way higher than you will ever achieve while doing the extraction it is not a problem at all. You will always be at the lower end of the saturation level and therefore the uptake will NOT be really lowered if you dont also stirr the organic layer, too.
So now maybe your initial concern was that you really want BOTH phases to mix completely, making them totally homogenous and then let them separate afterwards. Yes then it seems that you would need more stirring which is impossible.
But I highly recommend not to do it. The way you actually do it exactly the best IMO. Just stirr so hard, that both phases dont mix and the watery phase keeps separated, but always totally stirred.
Now all you have to do is wait longer, but even then its a benefit, as no emulsion is formed, which would take again like 20 minutes to separate fully.
My advice:
stirr it so hard that the phases just DONT mix (ok, it seems like you could not even get to this level, even if you wanted to) and stirr like 30 minutes.
Then you can easily decant the Naphtha which can be reduced and freeze-preciped directly, you wont need ANY additional workup, if you did not create any emulsions while stirring.
And place another portion of Naphtha on it and repeat.
Now one may think that NOT mixing the both layers together and let them separate from each other again is not that efficient. But with this method I only need 2 pulls to get 95 % of the soup, waiting 30 minutes or maybe even 45 for each pull.
So sorry for long post - but dont worry, your device is very good to go !!!