I would try using structured outputs recursively.

```

Request 1: "brown coffee table: " + {Root Schema} => "Furniture"

Request 2: "brown coffee table: Furniture / " + {Furniture Schema} => "Living Room Furniture"

Request 3: "brown coffee table: Furniture / Living Room Furniture / " + {Living Room Furniture Schema} => "Coffee Tables"

```

Many more round trips, but classifying products is not a latency sensitive task.