If you do want to go for a camera with a Full Frame sensor (the noise performance is worth it even against the best of the APS-C sensors), and given the budget you mentioned, then I would say the best option would be a Canon 6D or Nikon D600 or D610, depending on how you roll. BTW, that's not to say that their newer models aren't equal, just that they are considerably more than $1500 and not necessarily better or that much better (that will get me in trouble
). Used in good condition or possibly even as new-old-stock these should be available between $1000-1500. They compete extremely well on noise performance against the very best camera sensors available, including things like the new Sony A9 at $6-7K.
It is very illuminating (haha) and confronting to compare image quality in low light from a host of these cameras. DPreview (no affiliation) has a wonderful camera comparison widget you can use in every (most) of its camera reviews. It uses a studio image in normal or lower light. I choose a dark part of the image - my favourite is the dark area on the LHS of the wine bottle, set the image quality to RAW and Sensitivity to ISO 3200 (not that you'd use that, just that it amplifies differences / makes them more obvious) and the image illumination to low light. Set the image to COMP to compensate to equalise the image size on screen from the various FF/APS-C or other size sensors. Then look at the noise grain in the shadows and hot pixels etc...... It's an amazing leveller
I've attached a composite image I made from many of these comparisons. It contains a mixture of popular Full Frame and APS-C sized sensors. Unfortunately they changed to the image shown a few years ago, so one can't compare it with many of the other quality, but older models.
See what you think. Compare the $6-7000 Sony A9 or others .... with the others...
I accept there are other important camera attributes to consider, connectivity, image quality from available lenses etc, but low noise is certainly right up there.
Image attached - resize with "CTRL +" to study in detail.
I've also added an image of the entire studio low light scene used to give you an idea of scale. The highlighted area at the bottom centre of the images is the wine bottle dark area I used.
Best
JA