Currently everyone thinks that melee AD Carries are worthless because they blow up from huge CC. Why not add BKB? Because it would be broken as **** if every single champion was able to buy it; here is my proposal.
Some fancy name for magic immunity(Total cost of 3500)
+400 health + 35 damage
Unique Active:Shields your champion so it is immune to magic for 2 seconds(Only works for melee champions)
The active time limit and gold cost could be tinkered with testing and certain abilities like taunts/fear would be able to go through BKB seeing as a taunt/fear aren't exactly magic.
This wouldn't be super inherently broken because it lasts only two seconds, some forms of CC can still go through it making Olaf's ult not obsolete, and it would only work on melee auto attackers so ranged champions which are meant to melt will still melt with crappy positioning.
This is how DoTA solved the problem with Battle Fury(The dota version of Tiamat http://www.dota2wiki.com/wiki/Battle_Fury) Where a ranged champion would be extremely broken with an AoE splash of 225 range.
I honestly see no problems here other than teams actually needing to shutdown carries with constant ganks. Oh look suddenly melee AD carries like Yi are seen in the bottom lane.
BUT SIR YOU ARE COMPLETELY STUPID NUNU AND KAT WOULD BE SUPER BROKENZORED!! Well little Timmy, if you let Nunu or Katarina farm enough money for damage AND a 3500 gold item then you didn't shut them down at all and deserve to lose.
This post reminded me that one reason that melee carries work in Dota much better than they currently do in LoL is that many items are modified to be stronger on melee heroes. You mentioned that Battlefury's cleave only works on melee heroes. There's also vladimir's offering, which provides a lifesteal aura that only works for melee. Other items' effects are simply stronger on melees: cranium basher's bash chance is greater on melee heroes, Manta Style's illusions are stronger and the cooldown is shorter on melee heroes, Diffusal blade's mana burn only works on melee illusions, and so on. I understand that having different stats for items depending on whether a ranged or melee champ purchases it might be confusing or clumsy to handle, but it's something that might be considered for just a few items like the one suggested in the OP that would allow melee carries to be viable.