"Paparazzi" lens or camera change?

So I've bought the Panasonic G7 a while ago mainly for video and I was always satisfied. I had no problems with manual primes for most of my work and it served me okay. But my interest has shifted into photography and I feel like I can't get what I need with mft lenses.

I mainly take photos of artists before, during and after a music performance. I will try to explain my needs into categories:

Before the performance there are often photo lines, the lighting is good so I can mostly take any lens I want, you can get kinda near so even like a 30mm lens can work great, no big problems here.

During the performance is super dependent on the position. When I'm lucky and get the front row I only need a bit of zoom range because you can't really move around. But for larger stages you need 200mm+. The problem is the lighting, its often a bit darker esp during rehearsals. I have about 3 minutes time to take as many pictures as possible while getting shoved around by other people.

After the performance it's dark so you won't get any good lighting so you need a large aperture, have a very short time and need to be highly flexible with the zoom range and have to take pictures while moving.

The real problem for me seems that there is no large aperture lens with a good zoom range for the G7. On mft I can't raise the ISO to much and lenses seem kinda limited. The 40-150 olympus would probably be okay if it had OIS, but without body OIS it will be very useless for me. Also I feel like the buffer on my camera is full kinda fast, but maybe a better SD card could help out with that. Should I just look into switching to something like the Sony A7s series. Are there any lenses that could fix my problem?