People here often grumble when a questioner hasn't done enough work on their own before asking us to help. But in your case I think you've done too much!!
Bronte is doing something unusual here. Imagine you've been looking after a neighbour's child all morning and he has been driving you mad. When telling a friend about it later you might imitate the child's whine and say, "Why haven't you got any Coke? Why can't I go to Dweezil's house? When will Mum (Mom) be home?" Or you could put on exactly the same voice and say, "He wanted some Coke. He wanted to go to Dweezil's. When would his mum be home?" Have you heard that being done? It is a bit illogical: the child didn't actually say the words "He wanted" or "his mom". Nonetheless, I've certainly heard myself doing it.
So. The aunt didn't actually say "She regretted..." Bronte uses the speech marks (inverted commas) in such an unorthodox way because she wants the reader to imagine her aunt's voice.
What her aunt actually says is something like this: (You're probably way ahead of me and don't need the rest, but I'll do it anyway!)
"I regret being under the necessity of keeping you at a distance; but until I hear from Bessie, and can discover by my own observation, that your are endeavouring in good earnest to acquire a more sociable and childlike disposition, a more attractive and sprightly manner-- something lighter, franker, more natural, as it were--I really must exclude you from privileges intended only for contented, happy, little children."
A couple of things you queried:
to be under the necessity of: needing to or having to
endeavouring in good earnest: trying seriously
from privileges intended only for contented, happy, little children:
not "from things that only kids enjoy" but from things only children who act like children - proper children, in her view - deserve.
The aunt sounds pretty loathsome. As you say, she and Jane clearly dislike each other. I should read it.
Best Answer
So, your first suggestion seems apt - They are too busy to notice their own happiness.
This means that the happiest people are those individuals who really don't worry too much about their happiness. When you start noticing your happiness, you tend to measure it and we humans are never satisfied with what we have.
"I could be happier" - That is what creeps in when you think about how happy you are. When a person who is too busy to think about it doesn't waste time measuring his happiness. After all, Ignorance is bliss.