Hello John
John Samson is certainly listed in early 19th century trade directories as a silk manufacturer at that address, however I did limit the scope of my own study to the 18th century, and he doesn't seem to have been there at that time, the premises were occupied at that time by another silk manufacturer, Jean Michel Galiègue, a refugee to London with his father and brothers, from Wallincourt, Picardy in the 1760's.
Searched through all Peters listed on the land taxes in Spitalfields 1740-1760. Definitely no Peter Lalleman. Perhaps we are wrong to go down the route? Nearest listed is a Peter Lemon but would that have been transformed to Lawlaman? Not so sure.
There are some more references to the Peter Lalleman of Dublin on Ancestry, subscribing to some publications, he was married and he left a will, but everything links him to Dublin rather than London, so seems doubtful he is our man.
Peter Laleman, Male, 1738 Subscribed to 'A Political Essay upon Commerce' by Monsieur Melon, Translated, with some annotations, and remarks, (1738, BINDON, David. Dublin)
Peter Laleman, Male, 1738 Subscribed to 'A scriptural exposition of the Church-Catechism with devotions' (Fourth edn., 1738, HARRISON, Joseph. Dublin)
Peter Lalleman, Male, merchant in commerce, 1749, Subscribed to An essay towards an historical account of Irish coins, (1749, SIMON, James. Dublin)
Deaths 1779, Dublin 'In Pheonix Street, Mrs.Laleman, Lady of Peter Laleman, esq. formerly an eminent merchant'
Index to the Prerogative Wills of Ireland 1536-1810 : 1789 Peter Laleman, Dublin esq