Hi Robert,
A correct exposure will only get you so far.
A reflective object only has a dynamic range of about 4 stops - most monitors will display around 6 - and most cameras will capture around 12 (but only show the top 5 or 6 on the in-camera histogram).
This means that in essence "something has to give" ...
If you get a middle gray to expose in the middle if the histogram then a reflected white will be 2 stops above that - but - the cameras will be able to record about 1 stop above that point - so it won't be a pure white as far as the camera and monitor are concerned.
If you try to "get smart" and apply 1 stop of exposure compensation at time of capture so that the white is captured as a true highlight then - unfortunately - this also shifts the blacks up a stop as well, and what would have been a black - at 2 stops below middle gray - becomes only 1 stop (ie a reasonably dark gray, not a true black).
So if you're displaying on a monitor then you MUST stretch the tonal range of the image from the 4 stops (maximum) that cover the reflective object into the (typically) 6 stops that a monitor displays so that both your highlight & shadow points correspond.
If you don't then you'll have a tonal range offset, and that in turn will affect the perception of the colours.
The second issue you'll have is with colour calibration. It's great that you're working from a profiled monitor, but the accuracy you're seeing is only as accurate as the profile that converts the camera's capture into the image that's presented to the monitor. In other words, if you're serious about colour management then you need to get that Color Passport ASAP. Think of it as a profile to correct for the camera/conversion inaccuracies much like the monitor profile corrects for monitor/display adaptor inaccuracies.
The third issue is your lighting. Cameras have different response characteristics to the human eye - and using mixed lighting like you are is a bit like running your car on petrol ... with a little bit of diesel mixed in and a bit of oil mixed in for good luck too: In theory it'll handle it, but in practice, the performance won't be as good as if you stuck to just petrol.
Hope this helps