There are very few developers that will stay usable for more than, say, ten hours as a working solution. It sounds as though it was somehow becoming more alkaline, or active, before it oxidised and stopped working??
Unless there is a specification sheet saying differently, one-shot would be the normal way to use a paper developer. For a 5x7 dish that might only require 450ml or so of working solution, hence 45ml of the stock - so quite do-able, as you say.