The Spectator reviewer of that book in the above link points out that author Lucinda Hawksley presents no hard evidence at all for birth of any child. It's all speculation. Another reviewer I read and posted a link to here said the same.
Queen Victoria noticed everything and everybody and I don't think a daughter coming up to her eighth/ninth month would have escaped her notice! Princess Thyra, Alexandra's sister, did have a baby (the young father concerned shot himself after an interview with the King.) However, she was sent away in the latter months of pregnancy on 'a visit to her brother', the Greek King.)
Louise was at court the entire time she was supposedly pregnant. Victoria had had nine children, many of her her ladies in waiting had also produced large families. What are the odds of a very slender young girl getting away with a pregnancy under their eyes?
I read in a biography years ago too, that just before Louise's marriage Victoria wrote to the Duke of Argyll, Lorne's father, warning him that Louise was most unlikely to have children, was in fact barren. Whether this was the result of sparse periods or something else isn't known, but the Royal doctors seem to have been consulted about it and must have given a negative verdict about Louise's fertility.
I'm reading a biography of Prince Leopold at the moment by Charlotte Zeepvat. In it she describes Queen Victoria's reluctance to employ Stirling in the first place. He was a young army officer, not medically trained, (the start of his service coincided with the beginning of Leopold's epileptic fits which he suffered for the rest of his life.) Victoria didn't like smart young army officers who were used to London Society, but allowed herself to be persuaded into giving him a chance.
Leopold liked him, but Stirling soon blotted his copy book with Victoria by being fierce to one of her Highland servants. The man, called Robinson, acted as valet to Stirling but proved incompetent and was told off by his master. That was a sin of the deepest dye as far as Victoria was concerned. No-one spoke in a rude way to her Highlander servants, not even her family, and that, mixed with renewed worries about Leopold's health, (how would Stirling have managed an epileptic fit or fainting spell) meant he had to go.
Nothing to do with impregnating her daughter! Incidentally, during Stirling's brief term of service, Leopold, Arthur and Louise all came down with whooping cough. Very romantic!