If the environment being captured on a video stream becomes less complicated, does the bit rate and latency decrease? For example, a live GoPro video is being streamed to a HDMI capture software on Windows. The live stream of an office with the lights on has a bitrate of X and a latency of Y ms. If the lights in the office are turned off so all that is captured in the live video stream is a red light bulb and a black background, will the new bitrate be less than X? Will the new latency be less than Y ms? 

The resolution is maintained throughout at 1080p at 60fps.

I'd appreciate any help you folks can be!
so all that is captured in the live video stream is a red light bulb
  and a black background


No. You can't get black background. You will get any possible type of noises, many shadows and all kinds artifacts of camera's matrix.


  will the new bitrate be less than X?



No. As GoPro writes constant bitrate (CBR)
No. As new noises can require even more bitrate than static image.
Yes, if you will fill background with black color in video editing software and select non-CBR mode for compression



  Will the new latency be less than Y ms?


No. Delay causes by some buffers. Buffers for N frames, buffers for N bytes. If you lower bitrate, you can INCREASE delay in some cases (actually, it's true to very low bitrates).