How to properly clean raw chicken cut in pieces?
There are several studies published on the presence of campylobacter (which is the bacteria of concern) in chicken. You didn't specify where you are, but the findings are fairly consistent worldwide.  Chicken has a much higher incidence than other poultry products, and other proteins, though campylobacter can be present there too.  Of course, this is all made safe when cooked.  So, the concern with washing is cross-contamination, as you correctly point out.
I would use a paper or cloth towel to remove any bone fragments that might be dangerous if chewed or swallowed. Then dispose of the paper towel or wash the cloth thoroughly. Bits of fat can be removed this way as well, or cut off with a knife.
As for any remaining blood or parts that are not easily wiped away, just leave them.  Try it.  See if you think they make a difference in the final product.  I assume that you will not even notice them... or you can scrape that away after cooking.  If that is not possible for you, perhaps search out another butcher who will prepare it the way you like... or give your current a butcher another try, but this time ask him or her to remove the parts you don't like.
This all might be a bit of extra work, but you will soon find a rhythm... and, for the money you are saving, it might be well worth it.