Place: A Graveyard in Metropolis.
Bullseye has 5 hunting knives, 3 metal needles, and two hammers.
Azrael has his regular weapons.
Prep time: 2 weeks. Both are given resources.
Morals are off as usual.
Both want to kill.
Their attire is what they are wearing in the pics.
Who wins here?
I don't see how Azrael stomps a guy with no morals and inhuman accuracy. Bullseye can turn everything in the graveyard into a deadly projectile to harm Azrael with and his H2H ain't too shabby either. Even with all the advantages the OP gave Azrael this isn't a stomp at all.
Later on in his career azrael was lifting 7-8 tons and had meta human speed and agility well above Batman's. He also has "the system" programmed into his brain which kicks in at random times and allows him to do ridiculous stuff (kind of a deus ex machina sometimes) so I'd say he wins.