I had a similar problem. To get the best match to your BMP, I suggest doing what I did: convert to TIF or JPEG since avconv doesn't handle BMP very nicely - no matter many high quality settings I discovered and set.
A visual way to do this is simply get GIMP and BIMP (which is an excellent batch program anyway) and do a batch conversion of your BMPs to JPEGs at 100% quality or TIF (you could use TIF compression).
Now try the same avconv action. I used avconv -f image2 -r 15 -i animation%04d.jpg -r:v 15 -c:v h264 -crf 1 -an "animation_test.mov"avconv -f image2 -r 15 -i animation%04d.jpg -r:v 15 -c:v h264 -crf 1 -an "animation_test.mov"
but it should be fine with your mkv action now.