I'm confused; what you wrote seems to contradict itself:
Hmmm? The program we use, which we wrote. Requires that all uploaded paperwork is in .tiff 200 dpi or less. (...) I understand that .pdf may be better, but the customer dictates we upload in .tiff format.
So... your customers are demanding multi-page TIFFs? Or your customers who wish to do business with you are required to use your in-house program that requires multi-page TIFFs? I'm not understanding which is the case.
If it's the former, then I think you are stuck with printing to PDF first and converting the PDF to a multi-page TIFF. Apple has not built in this functionality of printing directly to multi-page TIFF, presumably because they cannot fathom anyone ever wanting such a thing.
If it's the latter, your company needs to get a better software designer.