Being listed on Dictionary.com doesn’t really prove anything because dictionaries are descriptive, not prescriptive. They are reference works to help people understand how words are used, whether correctly or incorrectly. (That’s why it made no sense for people to freak out when dictionaries started making note of the incorrect use of “literally” to mean “figuratively.” There was some strange idea that putting it into the dictionary legitimized it, when all dictionaries have ever done is record usage, both good and bad.)
The books don’t really prove anything, either. The author of the first one says he made up the word, and also defines it as being more or less synonymous with “arational.” And the second one is a self-published PDF that only uses the word in the title (though again, the content looks to be more about arationality). In both cases—assuming the second author isn’t just ignorant—it’s a deliberate coinage for a specific, rarefied purpose. There’s no pretense in either of it being an ordinary word.
And as @rojo pointed out, “unrational” violates the normal rules of English (since “rational” and its related words are of Latin origin). Searching into it further, I can only find any regular usage of the word in middle English, which was highly irregular and technically a separate language from what we speak today (i.e., modern English).
Note also that neither “rejecting ration” nor “lacking ration” make any sense as definitions because the “ration” of “rational” and “irrational” is a component, not its own word that is being modified. While there is an English word “ration,” it is not any part of “rational” and “irrational.” Instead, those words are using the Latin word for “reason.” So really, you would want to say “rejecting reason” and “lacking reason.”
But even that doesn’t work because you simply have a mistaken understanding of the connotation of each word. Even if we use the examples you have provided, “unrational” does not mean “rejecting reason.” Your examples make it out to be a synonym of “arational,” which means “not based on reason.” So “going with your gut” might be arational, even if it is not irrational (which means “against reason,” and would include “rejecting reason”). Really, if you want to mean “lacking reason,” then “arational” or “non-rational” would be more appropriate. But if you are trying to imply “rejecting reason,” then “irrational” is definitely the word you are looking for (no “ifs,” “ands,” or “buts” about it).