None of the suggestions you put forward, or that can be offered, I suspect will tick all your boxes. If you want good low light performance, you've got to be looking at the f/2.8 and wider aperture lenses. Perhaps even primes like 35mm, 50mm and 85mm, if you don't already own them. Zoom lenses that do offer those wide apertures and are within your budget like the old Sigma 24-70mm F2.8 IF EX DG HSM from 2008, are pretty lacklustre wide open and at the longer end of the range. It's definitely going to be a trade off somewhere along the line, either in terms of max aperture, vibration reduction or image quality.
Out of the the two you mention and the Sigma I mentioned, the Nikkor 24-120mm f/4 VR is the better choice in terms of IQ and the inclusion of VR for low light hand holding of the camera. Another option for a lens you might want to entertain is the old 35-70mm f/2.8 AF-D. Built like a tank and with decent image quality, it certainly would perform well in low light and delivers sharp images. The trade off is no VR and not as wide or long as the other options. These can be picked up for a lot under you budget.