Try Turning the headboard upside down, and start with one of the legs going up the stairs, and then see if it'll twist sideways and upwards. Hardwood would allow 1cm thick, but you want it as wide as possible to alllow you to get screws on the top and the bottom of the piece, that'll stop any rotation of it.
I already tried it that way up, it wont go. Yes some hardwood to match wold make it look less of a botch job.