Don't write a negative reference letter. Don't beat around the bush with fogging like "the position is not suitable for you". Set up a conversation where you tell him clearly and assertively what his negative trait is (you said impatience), and be constructive about how he can improve, and how to manage his career in the meantime (avoid or handle such situations). Do this very urgently, don't delay - it's already hurting his career. Do it verbally, behind a closed-door. Keep it constructive. Make it a two-way conversation, not a firing squad. Suggest or agree actions or metrics for the future.
One important duty of a supervisor is giving negative feedback. That means you. If you don't do that you're a bad supervisor and you're not serving him well. If you're consistently uncomfortable doing that with people, the issue is primarily with you, not him.
Do you mean:
- that this particular position requires an abnormal level of patience ('patience of Job'), or
- simply that he is in general abnormally or pathologically impatient?
Those are two utterly different scenarios warranting two different courses of action. You're not being clear. Or assertive. Impatience is potentially a very good trait for some positions (and bad for others), so do you really mean he lacks the social skills or communication style to mask his impatience? Really focus on being clear and specific. Was it foreseeable that 1a) he should have been able to figure out said position requires an abnormal level of patience (in which case, help him figure that out), or 1b) is it that you somehow know this via the grapevine and are trying to secretly "help him" without telling him why? (in which case, teach him how to do his own background check on a position)
I cannot turn down his request because he said he could not find anyone else. (He needs 3 references)
EDIT: based on discussion with @Corvus, here is a major cultural difference between academia and industry:
[In academia] References have a standard set of things, and it's considered ok to write a reference which intentionally omits some of those.
[In industry] Absolutely you can! In fact, arguably you're obligated to, ethically. Arguably, the moment you detected a sufficiently seriously negative personality trait that would harm his career under your supervision, you were obligated to tell him promptly - not delay until the last minute when it damages his career or livelihood - as it is now. There's a pair of you in this situation, as they say.
Tobias K.: "You may also want to contact the people you send the letter to and tell them that you did warn him that the letter would be bad"
This is all too weird and avoidant for words. If you're that unassertive and uncomfortable being a supervisor and giving essential feedback, you should step down immediately from being a supervisor, or at very minimum warn anyone when they start under you that you're incapable of giving negative feedback, and that their career will suffer for it. If people saw such a weird cover-your-ass but-I-told-him-so follow-up letter, they might conclude that the referee has basic issues supervising and communicating with people, and that the department is aware of this and doesn't care. Don't create that situation. Set up the conversation with him immediately. Don't be afraid of that conversation. Handled right, it may be the most important and constructive of his career. It may also equally be an important learning experience for you.
You may want to try the book/audiobook/course: "Crucial Conversations: Tools for Talking When Stakes Are High, Second Edition"