as Eric says “Alternatively you could check which printer they are using. Many printers support embedding watermarks automatically.”
unless they have a logo that is so light that it can be seen thru else it will print black.
you could always place it in a corner on everypage instead of a centered type watermark.
like on a page footer or page header (property printoneverypage=yes)
that’s about it, I don’t see any other miracle solutions