I suppose your "idiomatic" meaning of stay hungry comes from this page: https://idioms.thefreedictionary.com/hungry. I think this is not really an idiom but rather a logical construction from one of the alternative meanings of hungry listed at the bottom of the same page: "ambitious." In other words, in order to tell someone not to lose their sense of ambition, one could say "stay ambitious" or "stay hungry."
I would further consider this meaning of "hungry" to be a metaphor, that is, it represents a metaphorical hunger for success rather than a hunger for food.
In fact, just about any such phrase using the word "hungry" could have the same metaphorical meaning: in a context where "stay hungry" means "stay ambitious," telling a person to "remain hungry" or "keep yourself hungry" would have the same meaning.
In the context where the alternative is to eat food, however, you could use any of the following phrases to tell someone to remain in a state of hunger:
- stay hungry.
- remain hungry.
- be hungry.
The phrase keep hungry does not ring true to my ear. The word keep has various idiomatic uses, such as "keep calm", that have the form keep + (adjective), but this does not seem to be one of them.
The phrase go hungry is another idiom. It has a connotation of involuntary hunger:
If people go hungry, they do not have enough food to eat.
This does not seem to apply when food is available but one chooses not to eat it.
The definition of the phrase varies from source to source, however:
according to https://www.merriam-webster.com/dictionary/go%20hungry, going hungry is a chronic condition,
but https://idioms.thefreedictionary.com/go+hungry defines it as missing a single meal.
On the other hand, if the circumstances are that food has been set out to be eaten,
but it will be taken away soon, I would understand the phrase
eat something or go hungry
to mean if you do not eat food now, while it is available, you will suffer from not having enough food to eat in the near future.