Sadly, it doesn't pass muster with my inner art historian. (Although I don't work in the field, that's what my BFA is in, and I know a decent bit about clothing from that era.) The dress is decidedly not 1780s. We do in fact know something about clothing worn by girls during that period (we have a number of surviving portraits, for one thing), and a gown comparable to those worn by adults a decade later isn't really in the running unless Jane herself had a Tardis and could hop ahead for a look at what she'd be wearing to the local Assemblies once she was "out". ;)

The disproportion of head and body, BTW, isn't uncommon for portraiture of that period. It's usually seen in portraits done by the class of painter who travelled around the countryside painting the wives and children of local gentry (as opposed to the luminaries who had regular studios), and you find it in both Britain and America. Sometimes it comes from the head being added to a prepainted body, but when as here it's accompanied by other distortions of form (notice how short her upper torso appears to be) it's more likely to be the result of the painter's lack of skill or practice at portraying human anatomy.

Myself, I'd want to see the provenance of the picture itself. Oral tradition is notoriously iffy, and we already have some discredited tales from that source relating to Jane. ("The prettiest, silliest, most affected husband-hunting butterfly", for instance.)

It's a sweet face, though, and altogether a rather endearing picture. I can understand wanting it to be Jane!