Hello Greaves,
I am a descendant of Alice's brother George Thomas (b. 1874). I have been researching his family for sometime now and found your information about Alice to be extremely interesting. When I started to dig into the story a little further, I found a few newspaper transcriptions on BNA that may be of interest.
September 3 1898 - published in the Graphic, pg 5
"...RING PERFORMERS: the Genuine NEGRO KENTUCKY TROUPE of Singers and Dancers: Alice Daniels..."
I cant be certain which Alice this article is about, although I do believe that it is unlikely there was more than one performer who had this name (based off of census data and the number of newspaper search results). If this is her, it may explain the "foreign looking man" that she was living with in 1899. Based on adverts in the paper, she only performed with the troupe between August and September of that year.
In regard to the 1899 turpentine poising, I did find an article that claims this was not her first suicide attempt.
Oct 22 1899 - published in Lloyds Weekly News, pg 4 (and several other papers)
"... On the prisoner being now placed in the dock the police officer handed the magistrate a letter, and said that a month ago the prisoner was charged with attempting suicide by throwing herself over London Bridge".
The same article also goes on to offer a potential reason for why there is no mention of the child in future records.
"The gentleman who was caring for the child asked for bail in order that the prisoner might look after their child. This was refused, and the man then asked to be relieved of the child as he could not look after her properly, and the prisoner's female relatives absolutely refused to bare anything to do with the little girl. [...] the only course of action was for the child to be taken to the workhouse."
(To clarify, the article does say that this was not his child but that he was very fond of her and that there was an emotional display when the two were separated)
There are no details about the family's reason for not wanting custody, however its not hard to come up with a list of potential reasons.
A little fun fact to close with:
Alice's family must have had a musical gene of some sort, as I found a newspaper article about her brother George singing aboard his battalions troop ship during WW1.
-EM