I haven't been able to find a good source for his nationality from a quick search online, updates and corrections welcome.
Austrian, according to the AlpinWiki: https://www.alpinwiki.at/portal/navigation/erst-besteiger/erstbesteigerdetail.php?erstbesteiger=11137 and Austrian Alpine Club mountaineering group: https://www.gebirgsverein-services.at/bergsteigergruppe/ein-besonderer-vortragsabend-sepp-brunhuber-1904-bis-1988/
Thank you. I'd never heard of him so it was interesting to find out a bit about him. A sliding doors moment: he was camping at Grindenwald waiting for the weather to clear but had to leave (for work?) and Henrich Harrer took his place. Had Brunhuber been on the FA ascent of the Eiger, Harrer probably wouldn't have been invited to Nanga Parbat and would likely to have gone to the eastern front and met the same fate as Kasparek. No Seven Year in Tibet or The White Spider.