I shoot a lot of bands and musicians under stage lighting, but sometimes the stage lighting is in a constantly flashing or changing mode. I'd imagine that the types of shooting you're doing would be more like trying to get pictures of the crowd away from the stage lights, which is harder. I find that with my 5D3, with the performers themselves, just using the center focus point with 1/160 shutter speed works wonderfully. I run my lens wide open and put what I want it to focus on in the center and that works quite reliably, even when the performers are moving around. It breaks down when they're doing a lot of quick movement but backing off and getting a wider shot helps it a lot. A lot of my photos are facial close-ups of the performers while performing, and head and guitar shots, taken with the 70-200mm f/2.8L run wide open. The 1/160 freezes the performers unless there is something in fast motion and it gives that part a nice visible motion blur (drumsticks are a great example, and hair flying in a "headbanging" motion).
When there's not hardly any light hitting the people I'm wanting to photograph, the 70-200mm f/2.8L doesn't give good results and I often switch to the Sigma 85mm f/1.4. If it will autofocus, the results are a good bit better due to 4 times the light being captured. That's the problem, though -- if the light is too low, autofocus breaks down (it's slower anyway with that lens) and my vision isn't good enough to focus through the viewfinder (heck, I can't even see and detail what I'm shooting through the viewfinder if it's too dim to autofocus in a lot of cases). The 135 f/2L is a fast autofocusing lens and sometimes works well in dim light, and has allowed me to shoot in dim light and get decent results. In once case in a parking lot (outdoor show), it was the only lens that would work for me since I had no flash. The 50mm f/1.4 is another choice but it just can't seem to focus sharply in a reliable way with autofocus, plus it's a bit wider shot than I usually take. Take a lot of extras to be sure you get some in focus if you use that one.
The only good way to get good photos in your case may be to use the Speedlight flash units. If the venue is going to be dimly lit, like the venue I shoot in was for Halloween, I'll carry 2 Speedlights and the transmitter and bounce the light off the ceiling. The indirect light avoids redeye and harsh shadows and doesn't blind people so badly. If your venue doesn't tolerate flash, then you're out of luck with that approach. Just experiment and see what works best for you.