Credit card processing wasn’t the problem. Banking is the problem.

Talk to the processor, ask them which bank to use. ;). The client bank is rarely the problem, often it's the underwriting bank of processors.

What do you mean by "the processor" ?

Our website did not sell anything direct to consumer. Our "customers" were "advertisers" (affiliate programs that supplied us with content and either mailed us a physical commission check or pay out electronically). There was no payment processing of any sort. What we needed to do was clear physical checks and receive deposits from electronic / online banking services like PayPal and Paxum. We had a business credit card which we used to pay for things like web hosting, and we had 3 bank accounts: a chequing account, business savings and US currency. We did not have a merchant account.

It wasn't that the bank refused to process payments for us. They refused to hold our money and handle every day banking transactions like bill payments.