...in that time, even a seperated woman was a very scandalous thing indeed, and she would never have been accepted in the Elliot household, no matter her skills at toadying. For her to have been divorced is out of the question (due to the difficulty of obtaining one, the expense, the necessary act of parliament, etc).

No, the easiest, and most likely, explanation is that she is widowed and was left without the means to support herself - a very common occurrance. There's no need to go through convoluted theories to explain her status. If her situation had been other than the norm, JA would have said so, and Mrs Clay's presence in the Elliot's lives would have been given more weight and space in the story - if only from Anne and Lady Russell's argument against Mrs Clay's "low" and scandalous status. But it is not.