You are right about Brighton being noted as a "fun watering place" - it had a particularly racy reputation due to the influence of the Prince of Wales and his louche set of cronies. However, I'm sure Willoughby would have found plenty of opportunities for amusement in Bath. It is true that by the time of S&S it was no longer an exclusive resort for the ultra-fashionable at the top of the social scale. Through the 18th century, rather than simply dwindling into a place for elderly invalids, the social mix of the visitors broadened and the place went gradually down-market.

To illustrate this point, the link at the foot of this post is to a page with various contemporary accounts of Georgian Bath. Check out last couple of paragraphs of the excerpts from Tobias Smollett's novel "The Expedition of Humphrey Clinker", published 1771. Further down the page are excerpts from Pierce Egan's account of Bath in 1819 - the sections on the Sydney Gardens and the Upper and Lower rooms are particularly relevant.

Further illustration can be found on this page from the Jane Austen Society of Australia website, about prostitution in JA's world. The section about Bath starts about half way down the page:

http://www.jasa.net.au/london/prostitution.htm

Note: I've included this link to give some insight into what Bath was like. I am not suggesting that young Eliza was forced by necessity to become a prostitute like her mother - we don't know how she managed to survive between being abandoned by Willoughby and being rescued by Brandon.

As for the lack of chaperonage for Eliza and her friend in Bath, remember that the friend's father was generally confined to the house by ill health, and there was no one else to chaperone them. He must have thought that at fifteen or sixteen the two girls were too sensible and well brought up to get into trouble if allowed out together.