Catherine would not know or address her as Alice, so I hold that Alice is Eleanor's maid, who would travel with her to Lord Longtown's.
(I note that Lord Longtown is one of the people the General hoped to meet in Bath but did not.)
"A letter from my steward tells me that my presence is wanted at home; and being disappointed in my hope of seeing the Marquis of Longtown and General Courteney here, some of my very old friends, there is nothing to detain me longer in Bath." (Ch. 17)