I merely believe that he was not reasoning when making this mistake. That's all "unreasonable" means.
Given the rest of the story and his replies, I'm also inclined to believe that the mistake was ultimately due to lying to himself about the importance of bookkeeping and how fast funds should have been spent. But even absent that, "unreasonable" is a totally reasonable word for this.
No, the word for "not reasoning" is "unreasoning". "Unreasonable" means something else. In the phrase "reasonable mistake" I mean it in the sense of "ordinary or usual". E.g.: http://legal-dictionary.thefreedictionary.com/reasonable
It is not unusual for new or small businesses to just wing it on accounting for a while and then hire an accountant who has to tidy things up. (If you don't believe me, ask any small business accountant; my pals have story after story like this.) If we take him at his word, him thinking that the payroll service was handling the withholding is also reasonable. And him assuming everything was jake unless somebody was yelling at him is not just reasonable but typical of first-time entrepreneurs. The main unusual thing here is the scope of error and consequences, but startups are all about rapid increase in size, so that's not unusual in context.
Of course, this could also be fraud: he could have known for years that he wasn't paying withholding, hoping to make it up later. It could be that he actively avoided thinking it through or was willfully careless, which would make it negligence. But as he presents it, it's a reasonable mistake. You can see more about exactly what this means if you look in legal sources for "mistake of fact".
That said, I think your skepticism about his story is reasonable. That his partners have both quit after losing confidence in him would make me unsurprised if this turned out to be fraud, not just an honest but titanically stupid mistake.
Given the rest of the story and his replies, I'm also inclined to believe that the mistake was ultimately due to lying to himself about the importance of bookkeeping and how fast funds should have been spent. But even absent that, "unreasonable" is a totally reasonable word for this.