I have gone through the website in detail again, and as before, it doesn’t look good for finding an A4 printer with banner capability. The models named on the first page are the models K8600/K8600dn, for which the HP spec site shows that banner media type is not supported; models 710C and 720C which are 1998-2000 era printers no longer available; and models 7150, 7350, and 7550 which use Windows2000/XP drivers and which have no vendors supplying them in the UK (except for one apparently with refurbished stock.) I haven’t checked other brands since the last time I failed to find anything.
I guess we may have got some sort of result, just, but it does seem extremely surprising given the popularity of genealogy that there don’t seem to be more printers with this function easily findable, in the UK anyway.
I would be delighted if you have a better solution here.