You can pull and use the following classification details

for serialnumber: 97112455

"classification": [
                {
                    "international_code_total_no": "1",
                    "international_code": "008",
                    "us_code_total_no": "3",
                    "us_code": "",
                    "status_code": "6",
                    "first_use_anywhere_date": "2021-09-15",
                    "first_use_in_commerce_date": "2021-09-15",
                    "primary_code": "008"
                }
            ],

a full output of details is here.

To get the classification param you need to use the serial endpoints, and specify the extended fields you want to obtain.

For those who first need to do a trademark search, the process will be two fold, once search for the trademarks, get their serial_numbers, than pull it using one of the serial number endpoints.