Embed QrCode in fpdf in CMYK

Question

I used a fpdf class extension (http://www.fpdf.org/en/script/script44.php) to handle CMYK in the PDF, and all is fine: I can print my texts with 100% K in CMYK.

Then, I needed to generate a QrCode to be embedded into a PDF using fpdf.
I did it using PHPQrCode (http://phpqrcode.sourceforge.net/), and it works fine for my purposes.
All is fine on screen, but I need to send that PDF to a print shop to be professionally printed in CMYK.
I directed a book publishing company for 10 years so I am well aware of the "issues" with color handling, especially in this specific case in which the black RGB (0,0,0) absolutely DOES NOT match with the CMYK black (0,0,0,100). It will be converted to a very dark, but not fully black, 4-color-layers, and when printed, it will look both bad and usually nasty due to problems of "registro e retinatura" (those are technical printing terms of which I do not know the translation)

Now, my QrCode is in PNG, which is RGB, fine, I get it. But since the QrCode is actually just one bit, 0/1, white/black, isn't there a different file format that lets me specify the "black" as "black" in CMYK - that is 0,0,0,100?

From what I read, not even JPG is CMYK - it's something in between (JPG uses a color model named YCbCr. It is neither RGB nor CMYK.). It sounds like a video signal standard to me :(
If I cannot find a solution, I will leave the PNG with the RGB black: it will look bad, but hopefully using the highest error correction in the QrCode, it will still be readable :(