Well, got a Toon to BloodWing in BL 2 and tried Melee with both a Bladed Gun and without, and no it does not work for the Melee BAR Challenges.
And thinking about it, this is the conclusion I came to, the Melee BAR challenges will only progress per kill, so killing Bloodwing and meleeing him before picking up the mod for claptrap will not work because you are not “Killing” anything, unlike the “Capture a Guardian” mission in TPS where you are meleeing and killing the Guardians as fast as they spawn (side note; using a bladed gun for this counts for both “Melee” and “Melee w/Bladed Gun”).
Bloodwing does however count toward “Critical” Hits for any Gun that has a “Critical Hit BAR Challenge”, as for kills for any Gun, I believe those will not count for the same reason that I gave for Melee.
If anyone has notes to the contrary of what I have said please post, as I want the best information available for all.