A long shot but I am just wondering if the Daily News article could refer to Pizer. I know he was [officially] cleared by then but Mulberry st is just up the road from Berner st. Pizer thickset, Broad shouldered man could be a similar description? Plus he would have been coming from Mulberry st direction. Ill-treatment of a woman perhaps hinted at in the newspaper report, other evidence, foreigner. Maybe they decided to take another look at him and shadow him for a few days just in case the Policeman who gave him his alibi was mistaken say, about the exact time.
The article says the person is being 'shadowed'. This means his moves are being observed. The Batty St., lodger wasn't being observed. They pieced things together from witnesses.
What if Kozminski had rented a place to stage his attacks from before returning there and then just going around the corner to his actual home?
Evening News, Oct 17/88:
"On Monday afternoon the truth of the statement was given an unqualified denial by the detective officers immediately after its publication and this presumably because they were anxious to avoid a premature disclosure of facts of which they had been for some time cognisant. From the very morning of the murders, the police, it is stated, have had in their possession a shirt saturated with blood. Though they say nothing they are evidently convinced that it was left in a house in Batty-street by the assassin after he had finished his work. Having regard to the position of this particular house, its close proximity to the yard in Berner-street, where the crime was committed, and to the many intricate passages and alleys adjacent, the police theory has, in all probability, a basis of fact"
..and further on...
"With regards to statements current as to the finding of a blood-stained shirt at a lodging-house in Whitechapel, the Central News says the story is founded on some matters which occurred more than a fortnight ago. It appears that a man, apparently a foreigner, visited the house of a German laundress, at Batty-street, and left four shirts tied in a bundle to be washed. The bundle was not opened at the time, but when the shirts were afterwards taken out one was found considerably blood-stained. The woman communicated with the police, who placed the house under observation, detectives at the same time being lodged there, to arrest the man should he return".