Given the following references, it appears that batswoman is probably a typo or misspelling of batwoman, which is a woman employed to maintain a household (sweep, clean, fetch things, run errands).
A 1943 contemporary account of the fall of Tunis, The End in Africa, by Alan Moorehead, contains the following passage that implies a batwoman is a kind of household attendant:
We ran on back to Gafsa, where we had established ourselves in a comfortable Arab house. It even possessed a wood-burning bath-heater, which provided the only hot baths we were going to get for the next two months. Since luxuries were to be had in that pleasant place, I hired, in the absence of a batman, a batwoman. Hyah was without glamour. She was an aged and hideous Arab crone who swept the floors and handled the laundry.
Chambers's 1961 Dictionary gives the following definition of batman:
n. one who has charge of a bathorse: an officer's attendant
It gives as an inflected form:
For bathorse, Chambers's gives the following definition:
n. a pack-horse carrying an officer's baggage
A Dictionary of Slang and Unconventional English from 1953 tells us that a slang term for batman or batwoman is batty, relating to the Services, and that the slang term dates from about 1925.
The 1972 Supplement to the Oxford English Dictionary defines batwoman as "a member of one of the women's auxiliary services performing the duties of a batman." For batman, it says "Add: Now generally, an officers servant" and gives a source from 1755 (a letter to Colonel George Washington)! A 1955 source in the same entry says "Men employed as outside batmen in the married quarters were expected to clean and polish the houses, clean windows, cut lawns, fetch coal, and run errands."