If the customer is willing to compromise, I'd spot colour that in your situation.
Reduce to white underbase an 3 spots with half tones. You should be able to capture the design intent if you can manage your halftones (which you should be able to do on Process on a manual, anyway).
I can get at least 3 visibly distinct halftone levels from each screen, (when used for a fill), so that gives you at least 9 colours or shades of fill from your 3 remaining spot colours, plus your gradients.
I would do a mockup for them and say this is what you can offer. In my experience, most customers are pretty clueless. As long as the print follows the design intent, most are thrilled with whatever you print. (Although we do our best to excel and match mockups exactly, we've been given shirts from previous years by other companies that are awful and the customer has still been thrilled).