So you have four combinations. Did I understand correctly that things changed when you switched BB but didn’t change when you switched headphones?
Did you try all four combinations?
To me this sounds like both of the two BB have contact issues but on different channels (one on tip and the other on ring). If that’s the case then it’s very unfortunate 
I mean that I don’t know the statistics but I’d guess those kinds of manufacturing errors are rare - and to have two of them 