My idea is simple. The green money organization authorize the right to generate green money to the company. Then whenever the customers buy the goods from these companies, they can get a series of code that can withdraw money online. Then they can use the green money buy other green goods on the online store. So, the money does not appear in paper form, is another way to be green.
The example of $5 and $8 is a little bit bothering me. People may not buy the $8 carrot only because it is green or sustainable. Well, here I assume the green money and dollars are not exchangeable. And the way to make greenmoney to be attractive is that it has higher buying power. Or maybe it is the exclusive currency to buy some kind of green product.
I think the ratio could really help to measure the sustainability. In order to help create awareness among customers, the app could include a function to compute the probable ration with some major factors input by the users. This could be fun and also help to expand the ratio to the public.