Low light photography is one thing, but fast-paced, low light EVENT photography is another thing all together.
When you've got a paying customer and they are expecting to get hundreds of great photos from the evening, nothing but a 1dx or a 5d3 has reliable enough focussing in very dark conditions (amongst new cameras at least).
I've used a 60d for similar events, and its reliability of focus was nowhere near as good as the 1dx or 5d3 and it was often slow to react as I had to wait for it to try and make focus.
If you are expecting to do difficult professional work with your new camera, get a professional camera. If you go cheap, you can expect to not get paid if your photos aren't up to snuff.
Over the years, I've wasted a lot of money trying to spend less than I have to, only to spend more on the right stuff later on when I learned my lesson.
5D3's are cheap now, and it will still be a great camera in another 4 of 5 years. The 6D (while a good camera) is basically a 60d with a bigger sensor and is designed for amateur use. It's a lot cheaper than a 5d3 for a reason and that reason is in the qualities that you would only notice when you are really pushing your camera hard, like low light events and when you need the camera to act instantly and perfectly every time.