Just to be sure, is the issue there with the source as well, or is it being introduced by Premiere. If the latter, does it happen when you use certain effects (I saw you mentioning bw, so perhaps you're increasing the contrast as well).
I'm asking because if it really is the effect introducing the issue, then the best solution actually is finding a better way to do the effect.
Think of it with numbers: the original scene has numbers that go from -100 to +100. Think of -100 as black and +100 as white. Now, when you increase contrast you're saying: I want the difference between numbers to be bigger. With numbers you can do this by multiplying. So let's say, you multiply by 2. What used to be -50 (dark grey) becomes -100 (black) and what was +50 (light grey) becomes +100 (white). However, you can't go beyond -100 and +100, so black and whites stay the same. Eventually, the values towards black and white get clamped together, whereas the difference between different shades of grey becomes huge, much more noticeable than before. So what used to be a 100 difference between dark grey and light grey becomes a complete difference of 200 that goes from black to white. So yes, by increasing the contrast you can't avoid but introduce banding because you are introducing a huge difference between pixels that used to be closer. So if a gradient used to go like "1 2 3 4 5 6" it will then go like "4 8 12 16 20 24". The difference is much more visible and hence why you can see the banding.
What am I getting at with this? Well... if you want to increase the contrast of a clip to achieve a desired look, try to do it in a different way. For example, instead of increasing the contrast itself, try to use HLS colour correction to get the darker shades even darker and the brighter shades even brighter, or perhaps try to recolour by hand, maybe by manually overlaying a smooth gradient that goes exactly from the colour to the colour you want, and thus recreate the scene through some masks and so on. It is obviously more complex than just increasing the contrast of the clip, however if you want to keep the quality intact, it might be a necessary step to take.
OTOH, if the banding was originally there, well, I suggest you to try and use gradfun3, an avisynth filter part of the dither package. You can get the dither package here
. Just unzip the proper .dll and .avsi files into you avisynth autoload folder. Do check the filter's manual for more info on the settings and all.