You can't rely on the integrity of the ground/earth on the pipework of the sink. For all you know it could be connected to PVC pipework further down. In the U.K all pipework is meant to be cross-bonded which is then taken to the earthing bar. However in Japan I am unsure if this is what is done. Regardless you should be earthing/grounding it back to the incoming supply.
In a lot of countries that use two pronged devices they are actually fed from centre-tapped transformers or are double insulated devices. Either way they should not become dangerously live.
In short don't connect to the sink run it back to the earthing/grounding bar.
First things first, here's a link to Leviton's 7299 combination switch & GFCI instruction sheet. For a tamper proof it will be a T7299. The only reason I give Leviton is because I know the part number. Hubbell, GE or Cooper are just as good and make the same.
GFCI protection for both outlets.
Follow the instructions that come with the GFCI. There are leads for the switch and lugs for the GFCI and also the feedthru-protection of another receptacle.
No GFCI outlet behind the dishwasher (for easy resetting without dishwasher removal).
This is accomplished by feeding the dishwasher receptacle using the GFCI feedthru-protection lugs.
The switch only toggling the garbage disposal outlet and not the dishwasher outlet.
This is accomplished by using the leads on the combo switch & GFCI to feed the garbage disposal.
Am I going to need to run some more wire through the walls?
If there are no wires between the combo switch & GFCI then you will have to pull some romex between the two.
Should I investigate adding a GFCI breaker for that circuit?
The breaker will cost a lot more than the combo switch & GFCI. Also, if the GFCI trips then you have to go to the breaker to reset it. At least with the GFCI receptacle feeding your dishwasher receptacle, you will be closer.
What other solutions would achieve the same effect as listed above?
I think this is the way to go, so you don't have to pull your dishwasher out to reset the GFCI. The nice thing about this site is some of the people either can think out of the box or have faced this problem before.
Best Answer
Ok the GFCI would need to be accessible I usually put them in the cabinet between the sink and dishwasher. (My state did not adopt GFCI for dishwasher so if the customer wants them that’s where I locate them). Romex as the connection is not the correct method. A metal clad or flexible conduit to protect the wiring is the code compliant hard wire method. The Romex can get pinched moving the unit so the Romex stops at the old work box then metal clad to the junction box on the dishwasher. Yes almost everyone has a dishwasher wired this way but an inspector will red tag this. I have been called several times to change wiring like this. Since the dishwashers are fixed in place the max load of 50% of the branch circuit rating. Per NEC 210.23.A.2 so normally a dishwasher gets a dedicated circuit.