As a consultant speaking about the integrator its all about the relationships. If an integrator is responsive, demonstrates technical ability and solves his customer's problems, the customer will most likely put price as secondary. I have seen MANY times where i was told to use a specific integrator, even though the cost may be a bit more. This is not always a bad thing, but it can be bad, when the Integrator is leading the customer down a wrong path; or when the customer does not know (or care) enough to determine that he's getting "bad work". Now if the customer is installing his first system, it tends to be more about price. Once the integator's foot is in the door, then the critical determinator (is this a word?) is the relationship.
With Government bids, is may be a different story as cost plays the biggest role in determining the award.
Building value and establishing a trust relationship with the customer earns long-term business, especially in government work.
It's better to explain why your price is higher one time that to apolgize for poor quality later or not get invited back to build upon the system. Then again I don't "bid" many projects.
People want to do business with people they can trust. Having done excellent work for the client in the past is one way to establish trust, the other way is through referrals from mutual friends or relationships established through business or social organizations.
All that being said, your price still must be reasonable - you usually can't charge 2X or more than what the competition is charging without being able to offer a very good explanation. Keep in mind that, in most cases, the person making the purchasing decision is accountable to his employer and must be able to justify his vendor selection to the boss.
I have been involved in evaluating numerous security/surveillance proposals for clients. In private sector work, a case can usually always be made for selecting a higher priced contractor if he offers more experience, higher quality, or better overall value. But again, the price difference must be reasonable and justifiable. A difference of 10% to 30% can usually be justified, while a difference of 100% or 200% usually cannot.