You probably have one of two possible problems.
1) The disposer may be jammed. Check this by making sure the switch is OFF and the breaker is OFF. Insert the proper hex wrench into the bottom hex socket (center bottom of unit) and rotate it left and right to be sure it turns free and easy. If jammed, it may take a bit of forceful back and forth action to clear the jam. Look down inside the disposer with a flashlight to assure there is no obstructions. If all is clear, turn the breaker back on and try the unit.
2) If after doing step one and all is clear but unit still kicks the breaker, the disposer is probably faulty and may need to be replaced.
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
Assuming that there is no other electric device connected to that GFCI - also not via a downstream/"load" outlet:
There might be a safety issue with that disposal, so taking care of the danger of electricity is mandatory.
Connecting the disposal to another GFCI outlet via an extension cable could disclose whether the disposal or the GFCI is defect.
The disposal or metal sink or other conducting material in connection with the disposal should not be touched, i.e. it should be switched on and the last step should be to plug in the extension cord into another GFCI outlet. To stop this test, the first step should be to unconnect the plug of the extension cord.
After some years, the electric isolation of the disposal could get worse due to humidity, dust, vibrations, grease/oil, acid etc.
The test button of the GFCI might show no problem, but that button can not test an over-sensitivity.