] I have no idea as to whether Napoleon Bonaparte served in a similar regiment, but his family was at best minor nobility or so I understand. I do not think that four quarters (four noble grandparents) were needed.

Napoleon was an artilleryman. Artillery and engineers in the ancien regime army were not subject to the four quarters requirement. These were technical branches that a) required brains and b) weren't expected to engage in hand to hand combat like the cavalry and infantry. They lacked the social prestige of the traditional corps, and they needed to take their officers from among people with a certain amount of education. A bright Corsican landowner's son like Bounaparte, as he then was, would wind up there perforce.

Among the British also, gunners and sappers were not quite the thing socially. Even in Victorian times they had their own academy, Woolwich, while infantry and cavalry officers went through Sandhurst. Sapper officers, in the proverb of the British Army, were all "mad, married or Methodist," i.e. hopelessly bourgeois.