I don't know enough of the political workings of the HRE, but would they provide a way for Elizabeth to become a Stadholder?
Ehm, no, since the HRE has nothing to do with Stadhouders. A stadthouder used to be nothing more, nothing less than the governor of a province in the absence of the duke/count and appointed by him. (which is what it literally means, placeholder, i.e. lieutenant/steward) During the revolution the Provincial Estates started to appoint their own Stadhouders (i.e., each province for itself).
However, AFAIK Elizabeth was not offered Stadhouderschap (of individual provinces), but full sovereignty of the Netherlands. (presumably meaning: countess of Flanders, Duchess of Brabant, ...) This also means that it would be done on the sole authority of the Estates, and against the normal succession rules, so I doubt that the HRE would like it much!