The harpoon is inspired and I like the colours. Ideally the entire faction name rather than just BHG could be included, and a silver star could be incorporated for the BHG - representative of Texan origins and historic badge of bounty hunters.
The first BHG sig looks good. If it could have room for a character picture, name and possibly a rank (listed on faction status page) and maybe not have so many ships (or none?)
The harpoon thing could be adapted for the Core, but ideally the Core would be something a little more unusual, like a secret organisation with possibly a different colour scheme. The name is The Guild Core - bounty hunter part if kind of unimportant. Ships in Core sigs...well I'm just not sure if the BHG caps look all that good in sigs (the mako maybe). Like the background though.
Mottos are important as well. Problem would be finding a font that is characteristic enough.
