I only have 2005 NEC handy, but there's nothing there against using same circuit for both dishwasher and disposal.
210.23(A) Branch circuits, rated 20 or 15 amperes may feed lighting units, utilization equipment or combination of both. The rating of of any one cord and plug connected utilization equipment shall not exceed 80% of the branch circuit ampere rating.
Your local code could be different, of course.
Also note, that your garbage disposal needs a disconnect switch
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
I am assuming you are trying to install a switch leg for the disposal and a continuous hot for the dishwasher and the disposal and, dishwasher are on the same circuit, and you are trying to use a split duplex for both devices. I am also assuming that you are trying to surface mount the box inside the cabinet rather than use a cut-in box.
If you are installing a surface mounted box the conductors or cabling would have to protected no mater what kind of box you are using. If you came out of the side you would have to protect the conductors until you could pass into the interior of the wall or some other protected area.
One other problem exists the dishwasher circuit requires GFCI protection and the disposal doesn't. So if you do not have GFCI protection for this circuit you cannot put a split receptacle in under the cabinet. You would have to install two separate receptacles (at least one GFCI protected) or add a GFCI breaker to that circuit.
Taking this one step further. Since the NEC requires at least the dishwasher to be GFCI protected, it would be safe to assume that the area below the sink is considered by AHJ to be a wet location which would mean an "outdoor gang box" would be required in a surface mounted box.
FYI - The new 2020 NEC has modified 210.18 (A) which will require every 120V outlet to be GFCI protected including disposals, as well as any other wet location inside and outside a dwelling.
Hope this helps