I'm not sure
the child would have been considered a member of the burgher class... But it's true that most marriage between nobles and non-nobles were often considered morganatic and thus the children of such a union would have had no rights in the succession.
I wouldn't rule out the possibility of one getting the possibility to become King: we've had a few bastards that managed to achieve that (like John I of Portugal), so a child born from a morgantic union could also work. The thing is that I have no such examples that come to my mind...
Morganatic marriages. Hmm.
Edward V, Edward VI, Elizabeth I, Mary II and Anne, would, by continental standards be considered as the products of Morganantic Marriages. Even in Germany, King Leopold I of Baden was a product of such a union. So it is possible, if rare.