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.
This sounds highly unusual some plug mounted gfci s need to be reset whenever they receive power but as you said you can unplug it and plug it in without having to reset it so I would check the wiring at the switch to make sure that it is breaking the hot wire and not the neutral also make sure the bare ground wire is not contacting the neutral at the outlet or in the switch box.
Best Answer
There are some rare kitchen GFCI exceptions - e.g., refrigerator, but only when it is a single receptacle (not the usual duplex), but I doubt the downdraft would get an exemption because there is no real basis for that - as opposed to a refrigerator where there is a real risk of serious harm (both monetary and health) due to spoilage.
But more importantly, if it is tripping the GFCI then you very likely have an actual Ground Fault. Fix the problem, don't bypass it. I would be particularly concerned with a downdraft because things happen fast when cooking - spills & thrills & wet hands & steam - not a good place to have an actual ground fault within easy reach.