My cards for this year have arrived from the printers, and much to my relief, they've turned out fine, insofar as the image on them is actually legible and not too dark. This will be because I ordered them from BananaPrint instead of Moo.com and because I prayed furiously before placing the order. It's too close to Christmas to be doing trial and error with different printers now.
The images were exported from Clip Studio as tiffs with the "expression colour" set to CMYK in the TIFF Export Settings box. Then I used an online tool to convert them to pdfs as, despite insisting on images in CMYK colour, these printing websites don't accept tiffs.