My very first published picture book in rhyme had one near rhyme in it. BUT... it was so "near" that no one, including myself, ever noticed it until a year or so after the book had been published! Make sure you aren't allowing near rhymes to take the place of true rhymes throughout your story, and you can possibly sneak by with one or maybe two near rhymes. It will depend on how smoothly it reads to people who have NEVER SEEN your story before, and also how "invisible" the near rhyme is in the story.
I've spent as many as ELEVEN years on one picture book, to get my rhymes perfect enough in the story to "work." It's became much easier for me, after I learned to use a rhyming dictionary for my stories. When I come up with a word that won't rhyme with anything that makes sense in my story, I skim through my rhyming dictionary, looking for two words that rhyme that DO fit my story line. Then I build my verse around those two words, instead of trying to just find a rhyming word that will fit a word I've already thought of. (Did that make sense?)
For example, when I was writing a verse in my Covered Wagons Bumpy Trails book that needed to describe the rough weather conditions pioneers experienced when they were traveling across the country, I found the rhyming words, rain and terrain. Then I built this verse around those two words:
Thunder, lightning,
Floods of rain.
Mucky, muddy,
Wet terrain.
You CAN write your story with perfect rhymes if you work hard enough at it. Sometimes, you just have to write something a little differently than you had originally imagined it. But you can do it!
Good luck!