These have been many very good points by
@Clovishound and others for sure. To outline something in your budget I do have a Nikon Z50 and I have a test shot for you to evaluate. I would strongly recommend going for the newer Z50II as the advanced processor in the new model would help greatly with your use scenario. This example will give you an idea what you could expect.
Below are two examples of the same shot (one I did not use for the weekly contest) shot at ISO 16000 with my Z50 no flash or other lighting.
The first is exported from the raw file using Nikon's NX Studio this uses the camera settings (they are default) to process the photo in camera. Noise and sharpening and standard picture control used. This shows what the camera does.
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Here is the same photo using the raw file in Affinity V3 only adjustments used were tone control and noise then developed then minor sharpening that is all - to finish the photo I would do a few other edits bit wanted to target your requested reduction in noise. Note: the photos lighter due to my starting process if I would not have done tone the raw file would look very dark. I would add contrast and other edits if I were finishing it, I wanted to show noise control by camera and then software.
View attachment 426156
Basically the Affinity edit is the start of my processing both are 1200 pixels on the long side exported as .jpg the Nikon file is 706kbits in size the Affinity file is 1.49Mbits in size there are more available details using the raw file in an external editor using techniques to "preserve" them. I know they can not be created but they can be preserved.
I hope this helps - a Z50II with a two lens kit is around your budget and they are light and easy to pack.