Where can I bounty hunt in a BHG Bullhead?

[Scumbag]

Bounty Hunters Guild ID

Pilot carrying this ID is a Bounty Hunter, who:
Can be contracted to escort lawful ships and fulfill lawful bounty contracts.
Cannot ally with any unlawfuls.
Cannot use any transports with more than 4,200 cargo.

Allowed ships: Fighters, Bombers, Freighters, Transports, Gunboats.

Bounty Hunters Guild Core ID

Pilot carrying this ID is a member of the BHG Core who:
Can demand contraband, levy fines, and destroy ships if they refuse to comply with contraband demands, refuse to cooperate or are affiliated with a house at war with BHG Core in the Omicrons.
Can hunt pirates and terrorists within the Omicrons and Omegas.
Can freely engage Order and Nomad players.
Can demand nomad biologics.
Cannot participate in unlawful actions except as described above.
May not enter any system containing a jumpgate while flying a cruiser or larger.

Allowed ships: Fighters, Bombers, Freighters, Transports, Gunboats, Cruisers, Battleships

The Core does not hunt bounties. BHG does not fly capitals.

So you don't hunt bounties with a Bullhead. You have plenty other things you can do with it. Join the Core.