This is difficult, but I’d still associate the addr:housenumber with the street. It’s not clear what ‘Hamilton Cottages’ is, but it seems to be a themed set of buildings, a sort of micro hamlet. This is similar to a multi-story building, where the building address has a street address, and the building has a name.
The only way to record the Hamilton Cottages part would be a landuse = residential tag with a name - I’m not sure if this will work if it’s nested though.