I should think a treat reward would be fine and not make the bird poop more. Have you ever tried to force out something that isn't ready to come out? lol.
I would say there was more danger of your bird holding it in until it can reach a spot it is allowed to poop in which I'm sure if done frequently isn't very good for them at all, just like it wouldn't be for people.
Of course they do have different a different anatomy so what applies for us may not apply for them, what I do know is the urates need to be got rid of frequently (which is part of the reason birds poop so much, the other being type of food consumed)and some species are only designed to hold their poop overnight because their body adapts to be able to do so. Most people need to ''go'' several times during the day but can easily hold their bladder all night.
The latter theory, and it is only a theory cuz I've only heard peoples ideas for and against, Ive never seen any evidence, is what puts me off toilet training although a bird that got desperate would probably just go anyway.... and that said I wouldn't allow a dog or cat to poo where the hell it liked, but on the other hand they aren't designed to poop so often.
Anyhow I'm waaaaay off track now. My point was I think you will be fine to offer food rewards
