Thought I would come back and update this story. At last I have picked up Lavinia's trail!
The 1939 Register was the breakthrough, as I found only one person in Birmingham called Lavinia with her birthdate of 7 Dec 1877. This was a Lavinia Rooney. She seems to be alone in the household, so I didn't pay to open the record but worked back from there. I looked for a marriage, and found a marriage between Anthony Rooney and Lavinia Davis in 1925.
The electoral rolls showed Anthony and Lavinia Rooney living at 76 Wellington Street from 1920 onwards. Anthony died in 1933 and Lavinia married a George Thomas Lees in Q4 1940, then died in the same quarter.
I went back to a 1911 census return that I had kept. This was for a Lavinia Davis, brass polisher, purporting to have been married to Matthew Davis, a brass caster, for four years. I couldn't find a likely marriage so kept this on the back burner as "Matthew Davis" was the name of Lavinia's father and I smelt a rat! The address was 3 City View .... Wellington Street!
Ancestry has the Birmingham rate books so I looked up this address for 1912 to find that 3 City View was the same property as 76 Wellington St, and the occupier was ..... Anthony Rooney! So I had the slippery couple bang to rights.
In view of everything else, I think Lavinia must be the brass polisher in 1891 in Park Lane. Either she did not go to the reformatory after all, or it was another Lavinia Davis in the theft case. I haven't found another candidate, but as there are no details in the report it's a possibility.
I've still not found Lavinia in the 1901 census (Anthony was a private in the Warwickshire Regt. in barracks at Colchester). She is probably skulking under another name somewhere.