Again, I'll recommend checking out Ladybug and Baby Bug. They're magazines for really young readers. And, since they are so competetive and extremely picky about what they publish, I found it interesting to read what they had. I remember them being very short.
That's all I can think of. It does sound like there are a lot of restrictions, but it is a worthwhile thing. Approaching it from a language development perspective is interesting. But, again, it could make it difficult.
When I worked at a Tutor Time last year, I worked with beginning readers. They often read short stories that emphasized a sound, like you've described. The stories rarely had a rising action, climax, denouement--all that. There just wasn't time. Instead, they seemed to focus on a scene with some action in it. There was dialogue and a bit of humor. But it wasn't an entire story, just a moment in time. Maybe that would be easier. Also, it didn't have to rhyme all the time. It just stressed the sound with repetition, like cat, hat, fat, bat, etc.