Over here, we add incentive to healthy food by exempting it from the GST. There are no food stamps here, but perhaps something similar can be done to achieve a compromise.
For instance, have food classifications and different kind of foodstamps. Foodstamps under, for example, class A can only buy fresh fruit, vegetables and meat. Class B could be stuff like Bread, Eggs, etc. A higher classification foodstamp (like junk food "C") can substitute when buying class "B" or "A" products, but not the opposite. Finally, regulate the foodstamps so like 90% are class A or B, and the remainder are class C.
Just a thought...