From my 1906 map of Bootle, Barnton Street was between Webster Street and the Leeds & Liverpool Canal. It looks like it doesn't exist anymore, as a current map has St Elizabeth's RC Primary School at the same location. If you go to where the junction of Webster Street meets Blisworth Street, then Barnton Street was almost dead opposite. Hawthorne Tannery was on the opposite side of the canal to Barnton Street.
Looks like the name lived on as there is a Barnton Close on the other side of the canal on a current map.