We don't have the accounts either... However, focus on the SOURCE of the funds that flow through the account.
Essentially you should be obtaining information on THEIR customers almost as much as you would if THEIR customers were YOUR customers... The whole risk spectrum for 3rd party processors is similar to Attorneys in that they can act as arms length intermediaries for thier customers/clients. You want to close down that arms length...
Since their may be some reluctance to turn over customer information to you... the next best thing is to get all the information you can about the transaction activity of the 3rd party processor, who their customers are, and clamp a whole bunch of monitoring controls on their activity so that you are constantly assessing the activity that is flowing in and out of the account... Keep an eye on changes in activity or patterns of activity which are hallmarks of fraud. Where possible, request the ability to audit THEIR information about customers annually and as necessary... especially if they are doing ACH, use NACHA rules to help you as much as BSA rules.
Hope that helps a little.
Cheers!
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In life, there is a lot less that could get better and a lot more that could get worse.
MBA Fin/MBS HR
My views only!