I'm no longer an integrator and am glad to no longer be in these scenarios, but something you said baffled me a bit.
You said in the beginning of your post that you "honestly have been trying to find a new one." You then go on to say that the "old" integrator won the first two, and say "first one not by much".
If it wasn't by much, and you want a new integrator, then why did the first one win it? I can understand if the second bid happened first and the second company was WAY out of line (in terms of pricing), but if it was close, why not give the new guys a shot if you're unhappy with the old ones?
I know that as an integrator, if a client was up-front about the fact that they wanted another option, and then I was not awarded a project for being slightly more expensive -- that would have turned me off completely. I would not expect to win if I was dramatically more expensive, but if it's strictly a price-driven exercise, I would consider that indicative of why you're in the situation you're in now, i.e. "trying to find a new one."
Just something to consider. I obviously am getting the Cliff's Notes version and there could have been something else that transpired, but I ran into similar situations in a past life and never quite understood them.