I made a provisional workaround. I've created a class that extends "Laravel\Cashier\Invoice" and overrides the function "pdf(array $data)". In the controller, I use "$model->findInvoiceOrFail($invoice_id)" instead of "$model->downloadInvoice($invoice_id, array $data)". I then cast the returned instance to my custom class and then call "download()". This is far from optimal, but works for now.
I would be very happy if someone could give me a better solution.