What are effective strategies for negotiating with banks?

The best way to negotiate with a bank is to take a banker (or ex-banker along). To be able to negotiate (for whatever purpose), you need to have a solid understanding of the ecosystem. The banks honestly would not care much if they lose out on you as a potential client for something that is not making them money &/or is risky.

A banker would know the margins within which the bankers operate, their negotiating tactics and rules of engagement / disengagement and how much flexibility the bank would be willing to show in your case.