This is, judging by the lack of responses and my own incomprehension, clearly not a common word.
I imagine breeze-boards are porous in some way, such that they admit the passage of air ("breeze"). I managed to find a site which says beekeepers use breeze-boards to distribute chemicals, but I can't imagine how you would use cheesecloth as a desk. Maybe it is supposed to be ironic? Protesting college by making desks that you can't actually use for writing seems appropriate.
I've never heard of breeze board, to my knowledge. But I wonder if Ms Byatt is using it as a synonym for breeze block:
breeze block n (Miscellaneous Technologies / Building) a light building brick made from the ashes of coal, coke, etc., bonded together by cement and used esp for walls that bear relatively small loads Usual US names cinder blockclinker block