How high is the outdoor sink in relation to your gray tanks? It might not be feasible to drain the outdoor sink into a tank as it is too low.....
Just throwing it out there as I have never had an outdoor kitchen but the sinks look pretty low in the ones I've seen.
2014 Jayco Swift 281BHS, 300W Solar!
2015 F250 XLT 4x4 Crew Cab, Short box, 6.2 gas