Good thinking trying, atleast.
You don't seem too eager getting the project, because of the bad payment history and the client being cheap. I would keep the potential trouble to a minimum and stick to the first team with this one, unless it really damages your relationship with the second one (they should understand your position, though).
I don't know what standards you set for your business. Each person has to decide their own ways. However as I said, if you disclosing the information of bidding through both teams doesn't feel comfortable for you, it probably means that a company playing with integrity, wouldn't do it.
It's difficult to give better advice, given that I really don't know the details regarding history, relationships and arrangements with the different teams, your situation and acceptable conduct where you're located.