First, if you and the deadbeat customer are in the same state, I'd suggest you check your state's laws on small claims court. Some states allow small claims cases for as much as $25,000. Small claims courts are informal, and it's really easy and inexpensive to sue through them. Nolo Press has good information on how to sue in small claims court.
If that won't work for you, I'd suggest you find a business lawyer who litigates, and sit down with them. That will probably cost you $1-200 for a 30-60 minute consultation, and they can give you really good advice.
The lawyer's first suggestion will probably be to have them write a legal nastygram. That will probably cost you another $1-200, and there's an excellent chance you'll get your money quickly once the deadbeat customer knows a lawyer is involved. If that works, it would be $2-400 plus an hour or so of your time to collect the $10,000, which I'd consider a pretty good deal.
If that doesn't work, the lawyer can give you an estimate on the costs of a suit. You might be surprised at how inexpensive it can be.
It's very common for business-to-business sales contracts to have a clause saying that if you have to go to court, whoever loses has to pay the court costs and the winner's legal expenses. (If you don't have a clause like that in your sales contracts, you should add one.) Also, even without such a clause, your lawyer might be able to include a claim in a suit asking to recover legal and court costs, and if things are as open-and-shut as you say, you'd stand an excellent chance of getting them.
If you actually file suit, you're likely to get an offer to settle from the customer. There's a good chance they'll want to pay the original bill but not any costs related to the suit. So you should consider in advance what you might want to do if that happens.
A collection agency will pretty much do all that without you having to make much effort, but they'll take a good-sized percentage of anything they get. So you need to figure out if you'd rather do it yourself and get more money, or have them do it for you and get less.
I had a client who owed me over $10,000, but he was in another state, and the contract was with a shell corporation that had few assets. So I decided not to sue, not because the amount of money was so small, but because the chances of "piercing the corporate veil" and actually getting to the guy with the money were tiny. Even though the case was open-and-shut, I'd still likely have wound up with nothing.