Well firstly, greataxe, flail or whip?
I'm gonna go greataxe because despite the horrifically degrading stats it's likely the best option:
Choppyboi
M12"* WS2+* BS2+ S7 T7 W16 A6* 3+/5++
Melee: Sx2 AP-4 D2d6d1
Mindbulletbird
M12"* WS2+ BS2+ S6 T7 W16 A5 6+/4++
3 Psychics at +2*
Melee S:U AP-2 D3
So if he gets in before taking over 7 wounds and degrading his stats, he'd make 7 attacks, hit ~6, wound ~5 and then have half of them saved by a 4++- but the Melta Axe means he'd be doing ~10 damage anyway.
If the LoC brought the impossible robe relic (he did), that's a 3++ and only 7 successful wounds, which means a fully statted counterattack from the LoC.
The good news is that the LoC is quite terrible attacking back, with 5 attacks, 4 hits, just over one wound and just under one save, for 3 damage if he fails the 5+/5++
The only issue is the gigantic and ruthless amount of mortal wounds the birb would be outputting. At least two of his abilities cause d3 mortals, along with the smite also being the same. The Armor of Scorn relic gives one denial attempt, but when he's throwing things out with +2's you'd need a damn lucky roll even to get it off.
It mostly gets worse, because your only area outside of his threat is to be 12 + 18" away so that you can't be hit with the smites. That however is further than you can possibly charge, so you've neutered yourself.
My only advice would be to hide behind cover, but that won't work because bloodthirsters are big loud models with huge wings which are begging for mindbullets.
Essentially, the bloodthirster will probably take roughly four or five mortal wounds a turn, but will dominate in close combat- EXCEPT for the horrifically degrading stats which means that he'll start dropping off in the second turn and continue on like that. Both of them being flying means that the LoC will happily disengage every turn. Good news is that the BT gets his extra S and A every turn when he charges again, bad news is... yeah, mortal wounds.
(There's a ton of other things I've missed such as the locus of trickery and the tesla axe but those interact in a very weird way which is difficult to calculate)