Isn't BT considered by most to be a lossy format? Doesn't its data compression algorithm degrade clarity and resolution? At least thats been my understanding. if that is the case, then I think it (the data conversion through BT) could be one of the sources.
Depending on the bitrate of your MP3 files that could also be contributing.... or at the very least highly compressed MP3 is not going to help.
Not sure which iPhone you are using, but my iPhone 5c has a surprisingly decent sounding headphone out. It can't properly layer an image or portray the soundstage with my better cans. But it does play my K701 and HD650 PLENTY loud enough, and subjectively I think its definitely not boosting treble. So that seems counter-opposite to what you are experiencing. Its definitely not what I would consider a bright or harsh sounding circuit, and the K701 is not a very forgiving headphone with bad stuff upstream. Not sure what this means for its BT out though.
Unfortunately with those bluetooth all in one headsets, you've got BT compression, digital-analog conversion and amplification all on the same logic board inside the cans. So there's simply no way to determine what the exact cause is and get rid of it.
If it were me, I'd ditch BT all together and just strut around with the HD600 straight out the jack or amp that to your liking... it'll blow away your BT headset.
I am listeining to a lossy Dreamtheater www radio stream through the iPhone 5c => K701 as I type (no amp). its definitely not treble-harsh in any way.