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I also think that if the company has never done something like this before - they'd JUMP at the chance to work with you just for the fact that it'll open up a whole new revenue market for them.
A payroll processing service is the most convenient answer, but you still have to transmit the accounting data to them. In the case of adjustments, the information has to be transmitted and reflected in the next run. That generates additional paperwork.
It's safe to assume you've already got an accounting system in place. If you're cutting the checks by hand because your accounting system isn't functional for doing it, then you could just upgrade to better software. Whatever's done with payroll has to be included with General Ledger entries monthly anyway, and for all practical purposes it's better to have it on an integrated system. In the case of adjustments it could mean multiple General Journal entries a month.
As long as the data input is accurate, issuing checks should be totally computer generated, which can just as easily be done from your system as by a payroll service. The clerical part is breaking the checks apart after they're printed, getting the checks into envelopes, sealing the envelopes, running the envelopes through a postage meter and getting them mailed. There's office equipment to do most of that, and doing the job in-house could probably be completely handled by one bookkeeper.
The best bet would be to talk to your CPA about it. Some CPAs handle write-ups for client bookkeeping as a service, and some even handle payroll for their client companies on an outsourced basis.
This really has nothing to do with whether it's an affiliate or any other type of business, it's strictly an accounting decision that your accountant should definitely be a part of.