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.
Yes, that is the entire purpose of the load terminals of the GFCI. You can imagine that the protection element of the GFCI extends both to its receptacles, and then just imagine the load pair as another "receptacle".
As to the second part, yes, if you just connect the terminals together "before" the GFCI protection is in place, the GFCI will have no effect - remember, it can only sense problems on its own receptacles and its load terminals.
Best Answer
No, do not join all of the neutrals. The GFCI load terminals need the neutral to remain separate. Your diagram is correct in that respect.
However, kitchen outlets must be dedicated; the circuit may not be shared with lighting as you have in your diagram. This is almost certainly a code violation in any jurisdiction.