Well, if it's beta, you are already showing it to people, so I would definitely show it to someone with a high chance of being a customer.

If they don't like it, you can always email them / call them to have a second look when you feel you've made noticeable progress.

EDIT: If using it in production carries a lot of risk for this customer, don't pussyfoot around and advise him to wait.

Thanks, we are not yet public beta. We are still in private beta. But this customer is in need of solution, and want to try it out asap.

I'd say go for it if a.) the core functionality is intact and b.) you set the customer's expectations. I am in a similar situation at my company and we have begun letting customers use our product. My basic feeling is that the more people you can get using the product the better - they will provide you with feedback and help you test how things scale early on. If the only problem is small UI things, it shouldn't matter so long as you manage is expectations correctly.

I agree. If this guy is a customer instead of (friends, family, etc) for private beta, he's going to represent part of the people paying you for this.

Congratulations. You're basically getting early feedback by someone who really really wants this. He wants a beta to use in production, and he knows that. That's how much he wants it.

Lay down the ground rules as far as production support and bug fix turnarounds based on what you can manage without trying to spend every waking minute keeping them happy. They may not get every new feature request they ask for, and , like you said, some UI might change.