Was Barbara Mr. Slade's and Mrs. Ansley's?

The answer to your question is yes. Barbara is Mr. Slade's and Mrs. Ansley's daughter.
The author doesn't reveal this fact until the last line of the story. It certainly makes for a dramatic ending. Additionally, it also puts a dent in Mrs. Slade's (Alida's) smug attitude. The text reveals that Alida envied Mrs. Ansley's (Grace's) beauty and engaging way with men during their youth. Alida feared that she would lose her fiancé, Delphin, to Grace. So, she masqueraded as Delphin and wrote a letter inviting Grace to join Delphin at the Colosseum late one evening.

You do understand? I'd found out and I hated you, hated you. I knew you were in love with Delphin and I was afraid; afraid of you, of your quiet ways, your sweetness . . . your . . . well, I wanted you out of the way, that's all. Just for a few weeks; just till I was sure of him . . .

Of course, Alida's purpose was a diabolical one. Knowing that her friend had a delicate constitution, Alida hoped that Grace would take ill from the Roman fever. With Grace failing, Alida would have Delphin to herself.
However, the author makes a dramatic revelation toward the end of the story. Although Alida wrote the letter, Grace did meet with Delphin that night. The result of that one-night stand is, of course, Barbara. So, although Alida was married to Dephin for twenty-five years, she could not prevent Delphin from having an affair with Grace. The proof for her failure is Barbara, who is the daughter of Grace and Delphin. The revelation must have been a devastating one for Alida, who for twenty-five years believed that she had outsmarted her rival.