In order to calibrate data you take at 1x1 you have to have darks and biases that are also 1x1.
If your data is 2x2 you need to take biases and darks at 2x2 as well.
Flats are a little different. Technically you can resample flats (resize them) and use them if done properly. However, to keep the theme going, you can be safe and match the binning for the flats as well- then you will not have problems when using BPP.
So the statement "Take my lights at 1x1 and everything else at 2x2" is incorrect if we are talking about the calibration data (biases and darks). These you match.
I suspect what you are thinking of the creation of LRGB (or LRGBHa) type images for which, yes, one method is to bin the color information in order to minimize read noise.
However, this doesn't deal with your initial problem/error for BPP which is concerned with matching biases, darks, and flats to your images. There needs to be a 1 to 1 correspondence in binning. Since you do not have any binned 2x2 darks or biases... BPP says it can't calibrate your data.
-Adam