1) Stop working right now. Fuck the business continuity documents. You're more than double over your budgeted hours.

2) Take 40 minutes of your invaluable time and listen to Mike Monterio's talk, "Fuck you, Pay Me". https://vimeo.com/22053820

How the client will take it is relative to how it impacts you. The worse it is for you, the less it matters how it impacts a client. Ultimately, you are responsible for the position you're in, but the client is complicit. There's a good chance that you won't be able to use this client as a reference, and do not kid yourself by believing that you can "explain the situation" to future clients. It's time to cut your losses and move on.

You are currently working for job rate of $17/hr ($6000/350). As a consultant, you are underwater. It will take time to find more work, which must be factored in to your hourly rate. I used to compute average hourly working rate on an quarterly basis: total billings for the quarter divided by gross hours worked. The gross hours worked number included admin time, which I billed to my own company. You're cruising toward McDonald's wage if you don't turn this around.

I'm going to assume that this was a fixed-price contract, and that you included a budgeted number of hours in your contract. You need to draft a communication explaining that you've more than doubled the number of budgeted hours for the project, and that you cannot continue working unless the customer is willing to enter in to a new contract with additional billing.

If your original contract did not include a budgeted number of hours, then you're really in a bad spot. The best you can hope for is that you can walk away and not have the customer come after you for failure to deliver, and demanding a refund.

The key to all of this is communicating early, and communicating often.

Do not worry that the customer will be annoyed.

Do not worry that they won't like you.

Do not worry that you're asking for too much.

Tell the client that you've worked over twice the number of budgeted hours, and that you're not willing to go any further.

Tell them what your time is worth, and that you'll be happy to provide them with an estimate to complete the project.

When you deliver that estimate, make sure your rate is solid, and that the estimate includes a budgeted number of hours. Deliver a weekly report of the hours consumed and the work completed. Book the time required to track this information to an "admin" task. Include that item on your billing, but bill it at a $0 rate. Explain to the customer that this line item is the time required to track the progress of the project, and that you're such a nice guy, you're not going to bill them for it.

I wish you the best of luck, but I've been in this situation before as well. I've been worse. I've had a developer flake out, and I had to write a check back to a customer. This is consulting. You win some; you lose some. That's why consulting rates are $250/hr when wages are $50/hr. There are costs external to the customer's job that must be priced in. The customer's alternative is to hire someone (muy expensive), or to find some other schmuck to do the job on the cheap. The trick is not to be the schmuck.

EDIT: After submitting this, I realized that I didn't address your $60/hr billing rate. That number is insanely low for a consultant. If you're a rock star and you've got work lined up for months you can optimistically expect to hit 80% billable time. That is 80% of your 40 hour work week will be billable. Do the math:

80% of 40 hours is 32 hours a week of actual billable time.

On an annual basis, how many weeks of work do you think you can line up? My experience was around 3 out of every 4 weeks. That's 39 billable weeks a year.

With those numbers, you're on track to make around $75k a year. That's if you have a mature business with a good client book and return work. It took me three years to get there.

The other problem with a $60/hr billing rate is that you'll attract $60/hr customers. Otherwise known as people who will run you double-over your budgeted hours and demand that you keep going. Fuck those people. Double your billing rate and start selling solutions, not technology.

See also: https://news.ycombinator.com/item?id=6717875