Cool project.

Though if you are looking at QR codes for something like this (internal, on paper), strongly consider using data matrix codes. They are more compact, same available error correction codes, and actually have a composition story rather then spaces between them.

Do to how datamatrix codes can be composed, you could probably design a code that both fills the height and does a continuous stream of codes with no horizontal white space between them, just a solid bar as per the spec.