I'd like to take the time to say that Lt. Col. Sheppard is doing a fantastic job responding to community questions and concerns, and scans of the current listing compared to concerns brought up indicate that a lot of them have been worked out.
Now, onto the meat- as a competitor to the BHG, do not expect to be able to use their technology or bases. You appear to have addressed this in your current shiplist, good thinking. This does not mean that you are automatically hostile to the BHG, and considering their reach and power you should be careful about accepting contracts with criminals or the Order. At any rate, who your first contracts are is going to determine the entire shape of your future. As has already been pointed out, once you've made an enemy it will be quite difficult to get on their good side again.
The only concern I have in this manner is the potential of you switching sides- while there may be monetary advantages to doing so, you shouldn't even briefly consider it just because someone else offers a better rate. The hit to your credibility is not to be underestimated- think about what would happen if the Mandalorians took an offer from Kusari against the BAF. Would anyone consider them for long-term contracts, when they've already betrayed one? While people might still trust your efficiency for named, single contracts, you would be very hard-pressed to find an ally ever again. If you were to break a contract and switch sides, you shouldn't even consider an amount less than 500 million credits, and no offer should be accepted without adequate analysis of long term repercussions.
That said, you have the mindset and framework for a very respectable faction. So long as the imported name remains only a name (a la the Mandalorians in their current incarnation) that shouldn't be an issue- there's certainly precedent for it. I highly suggest that until you have a major contract, you refrain from taking any bounties against faction targets. If you keep your business as more of a military-for-hire rather than a simple bounty hunting organization, you have a better chance of finding a niche to flourish in.
I wish you luck, and heartily approve of your efforts.